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.