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 $F\left(\overrightarrow{w}\right)=\mu^2\left(\overrightarrow w\right)-K\sigma^2\left(\overrightarrow w\right).$