1. Suppose the interest rate is \(r=3 \%\) per year forever.

1. What should be, at time \(t=0\), the price of a security that pays 100 each year \(t \in\{1,2,3,4,5\} ?\)
2. What should be, at time \(t=0\), the price of a security that pays 100 in all year \(t \geq 6 ?\)
3. What should be, at time \(t=0\), the price of a perpetuity that pays 100 every year?
4. Explain the relationship between the security prices you calculated in questions (a), (b), and (c).

2. Consider a 4-year semi-annual coupon paying bond with an annual coupon rate of 5%, and a face value of $1000. Suppose the bond price is $1,100. Calculate the yield to maturity of the bond.

Problem: Suppose you run a pension fund and you have the following liability: you will have to pay retirees $1,000,000 in 15 years. Suppose interest rates are equal to \(1 \%\) forever and that there are only two bonds available in the market: a 2 year zero coupon bond, and a 20 year zero coupon bond.

1. What is the present value of your liability at \(t=0 ?\)
2. Suppose you start at \(t=0\) with an amount of cash equal to the present value of the liability. What portfolio of 2 year and 20 year zero coupon bond should you buy at \(t=0\) in order to be immunized against change in interest rates?
3. Suppose that the interest rate increases from \(1 \%\) to \(1.25 \%\) at \(t=0 .\) Suppose that you have bought the portfolio that you found in question (b). What is the approximate change in the value of your asset and liability? What is the exact change in the value of your asset and liability?
4. Re-do the calculation of question (c) assuming that, instead of the portfolio of question (b) you have bought a portfolio composed of 30 year bonds only. Explain the difference in results.

Problem: Suppose the yield to maturity on a one-year zero-coupon bond is \(8 \%\). The yield to maturity on a two-year zero-coupon bond is \(10 \%\). Answer the following questions (use annual compounding):

1. Is the term structure of interest rate upward slopping or downward slopping?
2. According to the Expectations Hypothesis, what is the expected one-year rate in the marketplace a year from now, \(E\left[r_{1,1}\right] ?\)
3. Consider a one-year risk-neutral investor who expects the yield to maturity on a one-year bond to equal \(r_{1,1}=6 \%\) next year. Should this investor buy a one-year or a two year zero coupon bond at \(t=0\) ?
4. If all investors behave like the investor in (c), what will happen to the equilibrium term structure according to the Expectations Hypothesis? Suppose for example that \(r_{0,1}\) stays the same. What will be the equilibrium value of \(r_{0,2}\) ? Will the term structure be upward slopping or downward slopping?

Question 5

Numerical exercise. In the file "BondQuotes-PS01-2021.csv" posted on the website

you will find quotes for US Treasury bonds published by the Wall Street Journal on

December 24th, 2020. All bonds chosen mature on February 15 th, in different years,

starting from 2021 until 2050 . Use the software \(\mathrm{R}\) for the following calculations. If

you need help for question (d), please wait for the TA section on Monday where more

detailed instructions will be provided.

1. What days of each year do these bond make their semi-annual payments?
2. What was the day of the last semi-annual payment for each bond?
3. What is the accrued interest and the "dirty price" for each bond? (take the
   "clean price" to be the mid point between the bid and the ask price.)
4. Assume that at the current time \(t\), which is December 24 th, 2020, the spot rate

is given by a third order polynomial:

\[r_{t, u}=a_{1} u+a_{2} u^{2}+a_{3} u^{3},\]

Price: $20.62

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