MTH 4500: Introductory Financial Mathematics

Midterm 1 (Practice 3)

Problem 1

The US dollar (USD) is a home currency for dealer \( A \), while euro is the home currency for dealer \( B \). The dealers \( A \) and \( B \) use the following rates for currency exchange: \[ \begin{array}{|c|l|l|}\hline \mbox{dealer }A & \mbox{buy}&\mbox{sell}\newline \hline \mbox{EUR } 1& \mbox{USD }4& \mbox{USD } 7\newline \hline \end{array} \quad\quad \begin{array}{|c|l|l|}\hline \mbox{dealer }B & \mbox{buy}&\mbox{sell}\newline \hline \mbox{USD } 1& \mbox{EUR }2& \mbox{EUR }9\newline \hline \end{array}. \] Prove that there is an opportunity for risk-free profit. You are allowed to assume that each currency can be borrowed without interest.

Problem 2

It is known that the interest is compounded continuously and that an investment of \( P \) dollars today will result in \( Q \) dollars in one year. What is the annual rate of continuously compounded interest in terms of \( P \) and \( Q \)?

Problem 3

Consider a bond with face value \( F \) that matures in \( 21 \) years and that pays coupons in the amount \( C \) at the end of years \( 2 \) and \( 5 \). What is the formula for the price of this bond if the continuous compounding rate is \( r \)?

Problem 4

Assume that there are two securities \( S_1 \) and \( S_2 \) whose initial prices at time \( 0 \) are \( S_1(0)=60 \) and \( S_2(0)=100 \). The prices at time \( 1 \) depend on the market scenario in the following way: \[ \begin{array}{|c|c|c|c|}\hline \mbox{Scenario} & \mbox{Probability}& S_1(1) & S_2(1) \newline \hline \omega_1& 0.3& 66 & 170\newline \hline \omega_2&0.7 & 42 & 70\newline \hline\end{array} \]

  • (a) Determine the covariance matrix between the returns of the given two securities.

  • (b) Determine the portfolio of the minimal variance that can be constructed using these two securities.

Problem 5

Assume that there are three risky securities each of which has the risk equal to \( 1 \). Assume that the covariance between each two of the given securities is non-negative. If the risk of the portfolio with weights \( \overrightarrow w=\left[\begin{array}{ccc} 20\% & 60\% & 20\%\end{array}\right] \) is equal to \( \frac{\sqrt{44}}{10} \), determine the covariance matrix between the returns of the given three securities.

Problem 6

Assume that the expected returns of \( n \) given securities are \( m=\left[\begin{array}{cccc}\mu_1&\mu_2&\dots&\mu_n\end{array}\right] \). Let \( C \) be the covariance matrix between the returns and assume that the sum of all components of the vector \( mC^{-1} \) is equal to \( 18 \). For each vector of weights \( w=[\begin{array}{cccc}w_1&w_2&\dots&w_n\end{array}] \) denote by \( \mu(w) \) and \( \sigma(w) \) the expected return and the risk of the portfolio whose weights are \( w_1 \), \( \dots \), \( w_n \). Find the vector of weights \( w \) that minimizes the function \( F(w)=\sigma^2(w) + 15\mu(w) \). We are assuming that all securities are risky (i.e. that all variances are non-zero) and that their covariance matrix is invertible.