Question

the vanguard group is one of the largest investment firms in the world with approximately $7 trillion in assets in 2021. vanguard offers investment funds in stocks, bonds, annuities, and many other forms. vanguard estimates the average correlation between the returns offered by stocks and bonds to be - 0.33 (vanguard research report, september 2021). suppose that a particular stock - index fund offered by vanguard has an expected return of 4.87% with a standard deviation of 18.47%, and an investment - grade bond fund offered by vanguard has an expected return of 5.28% with a standard deviation of 2.23%.
a. based on the information provided, what is the covariance between this stock index fund and the investment - grade bond fund? round your answer to four decimal places. if required enter negative values as negative numbers.
b. construct a portfolio that is 60% invested in the stock index fund and 40% invested in investment - grade bond fund. let x represent the stock index fund and y represent investment - grade bond. round your answers to one decimal place.
r = x + y
in percentage terms, what is the expected return and standard deviation for such a portfolio? round your answer for expected return to two decimal places and for standard deviation to four decimal places.
expected return %
standard deviation %
c. construct a portfolio that is 40% invested in the stock index fund and 60% invested in investment - grade bond fund. let x represent the stock index fund and y represent investment - grade bond. round your answers to one decimal place.
r = x + y
in percentage terms, what is the expected return and standard deviation for such a portfolio? round your answer for expected return to two decimal places and for standard deviation to four decimal places.
expected return %
standard deviation %

Explanation:

Step1: Recall the covariance - correlation formula

The formula for correlation $
ho_{xy}=\frac{\text{Cov}(x,y)}{\sigma_x\sigma_y}$, where $
ho_{xy}$ is the correlation between $x$ and $y$, $\text{Cov}(x,y)$ is the covariance between $x$ and $y$, $\sigma_x$ is the standard - deviation of $x$ and $\sigma_y$ is the standard - deviation of $y$. We know that $
ho_{xy}=- 0.33$, $\sigma_x = 18.47\%=0.1847$, and $\sigma_y = 2.23\%=0.0223$.

Step2: Solve for covariance

Rearranging the formula for correlation to solve for covariance gives $\text{Cov}(x,y)=
ho_{xy}\times\sigma_x\times\sigma_y$. Substitute the known values: $\text{Cov}(x,y)=-0.33\times0.1847\times0.0223=-0.001367\approx - 0.0014$.

Step3: Portfolio construction (part b)

The portfolio return $r$ is given by $r = w_xr_x+w_yr_y$, where $w_x = 0.6$, $w_y = 0.4$, $r_x = 4.87\%$, and $r_y = 5.28\%$. So $r=0.6\times4.87\% + 0.4\times5.28\%=2.922\%+2.112\% = 5.034\%\approx5.03\%$.
The formula for the variance of a two - asset portfolio is $\text{Var}(r)=w_x^2\sigma_x^2+w_y^2\sigma_y^2 + 2w_xw_y\text{Cov}(x,y)$. Substitute $w_x = 0.6$, $w_y = 0.4$, $\sigma_x = 0.1847$, $\sigma_y = 0.0223$, and $\text{Cov}(x,y)=-0.0014$:
\[

$$\begin{align*} \text{Var}(r)&=(0.6)^2\times(0.1847)^2+(0.4)^2\times(0.0223)^2+2\times0.6\times0.4\times(-0.0014)\\ &=0.36\times0.03411409 + 0.16\times0.00049729-0.000672\\ &=0.0122810724+0.0000795664 - 0.000672\\ &=0.0116886388 \end{align*}$$

\]
The standard deviation $\sigma=\sqrt{\text{Var}(r)}=\sqrt{0.0116886388}\approx0.1081 = 10.81\%$.

Step4: Portfolio construction (part c)

The portfolio return $r$ is given by $r = w_xr_x+w_yr_y$, where $w_x = 0.4$, $w_y = 0.6$, $r_x = 4.87\%$, and $r_y = 5.28\%$. So $r=0.4\times4.87\%+0.6\times5.28\% = 1.948\%+3.168\%=5.116\%\approx5.12\%$.
The formula for the variance of a two - asset portfolio is $\text{Var}(r)=w_x^2\sigma_x^2+w_y^2\sigma_y^2 + 2w_xw_y\text{Cov}(x,y)$. Substitute $w_x = 0.4$, $w_y = 0.6$, $\sigma_x = 0.1847$, $\sigma_y = 0.0223$, and $\text{Cov}(x,y)=-0.0014$:
\[

$$\begin{align*} \text{Var}(r)&=(0.4)^2\times(0.1847)^2+(0.6)^2\times(0.0223)^2+2\times0.4\times0.6\times(-0.0014)\\ &=0.16\times0.03411409+0.36\times0.00049729 - 0.000672\\ &=0.0054582544+0.0001790244-0.000672\\ &=0.0049652788 \end{align*}$$

\]
The standard deviation $\sigma=\sqrt{\text{Var}(r)}=\sqrt{0.0049652788}\approx0.0705 = 7.05\%$.

Answer:

a. - 0.0014
b. $r = 0.6x+0.4y$, Expected return: $5.03\%$, Standard deviation: $10.81\%$
c. $r = 0.4x+0.6y$, Expected return: $5.12\%$, Standard deviation: $7.05\%$