MTH 4500: Introductory Financial Mathematics

Midterm 1 (Practice 4)

Problem 1

The expected returns of three risky securities are \( \mu_1=10\% \), \( \mu_2=20\% \), and \( \mu_3=25\% \). What is the expected return of the portfolio whose weights are \( \overrightarrow w=\left[\begin{array}{ccc} 20\%& 50\% & 30\%\end{array}\right] \)?

Problem 2

The investor is allowed to buy or short sell the zero-coupon bond with face value \( {USD}105 \) and maturity \( T \). The price of the bond at time \( 0 \) is equal to \( {USD}92 \). The investor is also allowed to buy or short sell a stock whose price at time \( 0 \) is equal to \( {USD}92 \). In addition to the described transactions, the investor is allowed to take either short or long position in the forward contract on the underlying stock. The delivery price \( F(0,T) \) for the described forward contract at time \( T \) is equal to \( {USD}120 \). Prove that there is an arbitrage opportunity and explain how this arbitrage can be achieved.

Problem 3

Two risky securities \( A \) and \( B \) have returns \( K_A \) and \( K_B \). The return \( K_A \) satisfies \( \mathbb P\left(K_A=8\%\right)= 0.2 \) and \( \mathbb P\left(K_A=-2\%\right)=0.8 \).

The return \( K_B \) depends on the return \( K_A \). If \( K_A \) is equal to \( 8\% \), then \( K_B \) is always equal to \( 29\% \). However, conditioned on the event that \( \left\{K_A=-2\%\right\} \), the return \( K_B \) is either \( 1\% \) or \( 7\% \). The events \( \left\{K_B=1\%\right\} \) and \( \left\{K_B=7\%\right\} \) are equally likely.

Determine the covariance \( \mbox{cov}\left(K_A,K_B\right) \) between \( K_A \) and \( K_B \).

Problem 4

One risk-free security and \( n \) risky securities are available in the market. It is known that \( P_1 \) and \( P_2 \) are two portfolios on the capital market line. The risks and expected returns of these two portfolios are \[ \left(\sigma_{P_1},\mu_{P_1}\right)=\left( 40\%, 21\%\right)\;\mbox{and}\;\left(\sigma_{P_2},\mu_{P_2}\right)=\left(70\%,33\%\right).\] What is the return on the risk-free security?

Problem 5

Assume that there are \(n\) risky securities and that \(\overrightarrow m\) is the vector of their expected returns. The covariance matrix between the securities is not known to us. If \(\overrightarrow a\) and \(\overrightarrow b\) are the vectors such that for each \(\mu\) the minimal variance portfolio with expected return \(\mu\) is given by \(\overrightarrow{w}_{\mu}=\mu\overrightarrow a+\overrightarrow b\), prove that \(\overrightarrow a\overrightarrow m^T=1\) and \(\overrightarrow b\overrightarrow m^T=0\).

Problem 6

Assume that there are \( n \) risky securities whose covariance matrix is \( C \) and the vector of expected returns is \( \overrightarrow m \). For every vector of weights \(\overrightarrow w\) let us denote by \(\mu\left(\overrightarrow w\right)\) and \(\sigma\left(\overrightarrow w\right)\) the expected return and the risk of the portfolio with weights \(\overrightarrow w\).

Assume that \(K\) is a real number such that \(K > \overrightarrow mC^{-1}\overrightarrow m^T\) and the matrix \(N=\left[\begin{array}{cc} \overrightarrow mC^{-1}\overrightarrow m^T-K & \overrightarrow uC^{-1}\overrightarrow m^T\newline \overrightarrow mC^{-1}\overrightarrow u^T & uC^{-1}\overrightarrow u^T\end{array}\right]\) is invertible. Here \(\overrightarrow u\) denotes the vector whose all components are \(1\).

Determine \(\overrightarrow{w}\) that maximizes the function \[F\left(\overrightarrow{w}\right)=\mu^2\left(\overrightarrow w\right)-K\sigma^2\left(\overrightarrow w\right).\]