MTH 4500: Introductory Financial Mathematics
# Midterm 1 (Practice 5)

**Problem 1**
**Problem 2**
**Problem 3**
**Problem 4**
**Problem 5**
**Problem 6**

The prices of two securities at time \( 0 \) are \( S_1(0)=100 \) and \( S_2(0)=120 \). The vector of their expected returns is \( \overrightarrow m=[0.3,0.8] \). A portfolio consists of \( 3 \) shares of the first security and \( 5 \) shares of the second security. Determine the weights of the portfolio and the expected return of the portfolio.

Consider a bond with face value \( F \) that matures in \( 30 \) years and that pays coupons in the amount \( C \) at the end of years \( 4 \) and \( 7 \). What is the formula for the price of this bond if the continuous compounding rate is \( r \)?

The interest rate is \( r=10\% \) and the interest is compounded annually. The coupon bond has face value \( F={USD}484 \), maturity \( T=2 \) years, and pays a coupon \( C={USD}44 \) at the end of the first year. Only the face value (and no coupon) is paid at maturity. If the price of the bond is \( P={USD}438 \), prove that there is an arbitrage opportunity and describe how this arbitrage can be achieved.

It is assumed that the investors are allowed to borrow and invest at rate \( 10\% \) whenever, as often, and as much as they want. Annual compounding means that if someone borrows the amount \( X \), the person owes \( X(1+r) \) after one year, and \( X(1+r)^2 \) after two years.

There are \( 5 \) risky securities with covariance matrix \( C \). The entries of \( C \) are not known. However, it is known that the portfolio with weights \( \overrightarrow{w_0}=\left[\begin{array}{ccccc} 0.2& 0.3& -0.5& 0.3& {0.7}\end{array}\right] \) has risk equal to \( 0.2 \). Assume that \( \overrightarrow{z}=\left[\begin{array}{ccccc} 3& -2&1& -3& 2\end{array}\right] \) and that the function \( F \) is defined by \[ F\left(\overrightarrow{w}\right)=\left(\overrightarrow{w}C\overrightarrow{w}^T\right)\cdot \left(\overrightarrow{w}\overrightarrow{z}^T\right).\] Determine \( \nabla F\left(\overrightarrow{w_0}\right) \).

The risk free rate is \( R=10\% \) and there are 4 risky securities whose expected returns are \[ \left(\mu_1,\mu_2,\mu_3,\mu_4\right)=\left(30\%, 40\%, 20\%, 70\%\right).\] The covariance matrix between the returns is not known. The expected return of the market portfolio is \( 90\% \) and the risk of the market portfolio is \( 40\% \). An investor holds the portfolio with weights \( (10\%, 30\%, 40\%, 20\%) \) and the investor has decided to change the investment to a new portfolio that has the same expected return as the old portfolio and that, unlike the old portfolio, includes the risk-free asset. What is the minimal possible risk that the investor can achieve?

Assume that there are \( n \) risky securities whose covariance matrix is \( C \) and the vector of expected returns is \( \overrightarrow m \). Assume that there is a risk-free security with return \( R \). Assume that \( \sigma_0 \) is a given positive real number. Determine the portfolio whose risk is \( \sigma_0 \) and the expected return is the highest possible.