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Rob's s1452 Notes

✏️ Examples (Risky Portfolio)

Name
Equation
Expected Return of 2+ risky assets
E(rp)=w1E(r1)+w2E(r2)...E(r_p) = w_1E(r_1) + w_2E(r_2) ...
Ex: Erp = 60%*11% + 40%*7% = 9.4%
Variance with 2 risky assets from Covariance
Var(rp)=w12σ12+w22σ22+2w1w2Cov1,2Var(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(rp)=w12σ12+w22σ22+2w1w2σ1σ2Corr1,2Var(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).
σP=Var=σ_P=\sqrt{Var}= 0.010692^.5 = 10.34%
If you have a target E(rPE(r_P)
w1=(E(rp)−E(r2))(E(r1)−E(r2))w2=1−w1w_1 = \frac{(E(r_p) - E(r_2))}{(E(r_1) - E(r_2))} \qquad w_2 = 1 - w_1
Correlation Coefficient (ρ1,2)(ρ_{1,2})
ρ1,2=Cov(r1,r2)σ1σ2=0.002667%×19%=.2ρ_{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
Cov(r1,r2)=ρ1,2σ1σ2=.2×7%×19%=0.00266\begin{aligned}Cov(r_1,r_2) &= ρ_{1,2}σ_1σ_2 \\ &= .2 × 7\% × 19\% = 0.00266\end{aligned}
Def of Covariance =E{[r1−E(r1)][r2−E(r2)]}=E\{\lbrack r_1-E(r_1)\rbrack\lbrack r_2-E(r_2)\rbrack\}
Utility Function
U=E(r)−12Aσ2U=E(r) - \frac{1}{2}Aσ^2

✏️ Consider the following risky and risk-free assets:

E(r)σCorrelation w/ Bond
T-Bill2.40%00 (risk free)
Bond fund4.00%9%1
Stock Fund10.00%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 E(rp)=(Er1×w1)+(Er2×w2)=60%×10%+40%×4%=7.6%\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 Var(p)=σp2=σ12w12+σ22w22+2w1w2σ1σ2ρ(1,2)=(9%2×40%2)+(15%2×60%2)+(2×40%×60%×9%×15%×−0.221)=0.007964\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}Standard Deviation =(0.007964).5=0.089241=8.9%\begin{aligned} \text{Standard Deviation } &= (0.007964)^{.5} \\ &=0.089241 \\ &= 8.9\% \end{aligned}

Another way to calculate it:

SD=SQRT(9%2×40%2+15%2×60%2)+2×40%×60%×9%×15%×(−0.221)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 Cov(r1,r2)=σ1×σ2×ρ1,2=9%×15%×0.221=−0.0029835\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}Var(p)=σp2=σ12w12+σ22w22+2w1w2Cov1,2=(9%2×40%2)+(15%2×60%2)+(2×40%×60%×−0.0029835)=0.00796392\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}SD(p)=0.00796392.5=0.0892408\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)E(r)sdsd
T-Bill2.40%0
60/40 risky portfolio7.6%8.9%
✔ Click here to view answer Erc=rF+y(ErP−rF)=2.4%+85%(7.6%−2.4%)=6.82%\begin{aligned} Er_c &= r_F + y(Er_P-r_F) \\ &= 2.4\% + 85\% (7.6\%-2.4\%) \\ &= 6.82\% \end{aligned}σc=σp×y=8.9%×85%=7.565%\begin{aligned} σ_c &= σ_p \times y \\ &= 8.9\% \times 85\% \\ &= 7.565\% \end{aligned}Var=σ2=7.565%2=0.0057\begin{aligned} Var &= σ^2 \\ &= 7.565\%^2 \\ &= 0.0057 \end{aligned}Sharpe Ratio =Er−rfσ=6.82%−2.4%7.565%=0.58\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=2A=2, what would your utility be, using our standard formula for utility U=E(r)−12Aσ2U = E(r) - \frac{1}{2}Aσ^2?

✔ Click here to view answer U=E(r)−12Aσ2=6.82%−.5×2×0.0057=0.0625\begin{aligned} U &= E(r)- \frac{1}{2}Aσ^2 \\ &= 6.82\% - .5 \times 2 \times 0.0057 \\ &= 0.0625 \end{aligned}U=E(r)−12Aσ2=6.82%−.5×2×7.565%2=0.0625\begin{aligned} U &= E(r) - \frac{1}{2}Aσ^2 \\ &= 6.82\% -.5 \times 2 \times 7.565\%^2 \\ &= 0.0625 \end{aligned}

Suppose that bonds earn 6% and stocks earn 10%. You want your risky portfolio to earn 9%. What must the weights on stocks and bonds be for the return on the risky portfolio to be 9%?

✔ Click here to view answer

We’ll use the following equation:

w1=(E(rp)−E(r2))(E(r1)−E(r2))w_1=\frac{(E(r_p)-E(r_2))}{(E(r_1)-E(r_2))}

and w2=1−w1w_2=1-w_1.

We’ll take bonds as asset 1 and stocks as asset 2.

w1=9%−10%6%−10%=−1%−4%=25%w_1=\frac{9\%-10\%}{6\%-10\%}=\frac{-1\%}{-4\%}=25\%

Therefore, the portfolio should be 25% stocks and w2=1−w1=1−25%=75%w_2=1-w_1=1-25\%=75\% stocks.

Let’s confirm this works. The expected return on the risky portfolio will be:

E(rp)=w1×E(r1)+w2×E(r2)=25%×6%+75%×10%=9%E(r_p)\\ =w_1×E(r_1)+w_2×E(r_2)\\ =25\%×6\%+75\%×10\%\\ =9\%