MTH 4500: Introductory Financial Mathematics

# Midterm 1 (Practice 5)

Problem 1

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.

Problem 2

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 $$r$$?

Problem 3

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.

Problem 4

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 $$\nabla F\left(\overrightarrow{w_0}\right)$$.

Problem 5

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?

Problem 6

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.