1.1 1.6.1

Suppose you buy 500 shares of stock A, buy 300 shares of stock B, and short sell 100 shares of stock C. The corresponding prices of shares A, B, C are $70, $50, $90. Compute the weights of your portfolio.

Solution. The amount of money invested in each stock is

\[\left\{\begin{matrix}X_A=500\cdot70=35000\\X_B=300\cdot50=15000\\X_C=-100\cdot90=-9000\end{matrix}\right.\]

so the total money invested is

\[X=X_A+X_B+X_C=41000\]

which yields the weights as

\[\omega=\left(\frac{X_A}{X},\frac{X_B}{X},\frac{X_C}{X}\right)\approx(0.854,0.366,-0.22).\]

1.2 1.6.2

Suppose you want to invest $5000 in 3 stocks A, B, C with the corresponding prices of shares $40, $50, $10. The weights of your investment to A, B, C are −0.2, 0.5, 0.7. Explain how to build the desired portfolio.

Solution. The share count for each stock is

\[\left\{\begin{matrix}n_A=5000\cdot(-0.2)/40=-25\\n_B=5000\cdot0.5/50=50\\n_C=5000\cdot0.7/10=350\end{matrix}\right.\]

thus we should sell 25 shares of stock A, buy 50 shares of stock B and buy 350 shares of stock C.

1.3 2.3.1

Consider a portfolio including stock shares of Jazz, Inc., Classical, Inc., and Rock, Inc as follows.

Compute the rate of return \(r\) of the portfolio.

Solution. The portfolio return is

\[r_P=17\%\cdot\frac{1}{5}+13\%\cdot\frac{4}{5}=13.8\%.\]

1.4 2.3.2

Consider a portfolio of two assets A, B with corresponding weights \(\omega_A,\omega_B,\) returns \(r_A,r_B,\) volatilities \(\sigma_A,\sigma_B\) and correlation \(\rho_{AB}.\)

1. Compute the mean and variance of the portfolio return.

2. Show that the portfolio return volatility is less than \(\omega_A\sigma_A+\omega_B\sigma_B\) for any \(\omega>0.\)

Solution.

1. The portfolio return mean and variance are

\[\left\{\begin{matrix}r_P=\omega_Ar_A+\omega_Br_B\\\sigma_P^2=\omega_A^2\sigma_A^2+\omega_B^2\sigma_B^2+2\omega_A\omega_B\rho_{AB}\sigma_A\sigma_B\end{matrix}\right.\]

2. Since \(\rho_{AB}\leq1,\)

\[\sigma_P\leq\sqrt{(\omega_A\sigma_A+\omega_B\sigma_B)^2}=\omega_A\sigma_A+\omega_B\sigma_B\]

as desired.

1.5 2.3.3

Consider a two-asset portfolio with \(r=(0.12,0.15),\sigma=(0.2,0.18),\omega=(0.25,0.75)\) and \(\rho_{12}=0.3.\) Compute the mean and variance of the portfolio return.

Solution. Calculations give

\[r_P=0.25\cdot0.12+0.75\cdot0.15=0.1425\]

and

\[\sigma_P^2=0.25^2\cdot0.2^2+0.75^2\cdot0.18^2+2\cdot0.25\cdot0.75\cdot0.3\cdot0.2\cdot0.18=0.024775.\]