**Objective**

The objective of this learning module is to establish expertise in the valuation of different types of fixed income instruments. The first few problems reinforce your knowledge of discounting cash flows. Embedded in these questions is the concept of risk identification.

In question 1, we determine the impact of a surprise in the interest rates.

In question 2, we calculate the modified duration and elasticity of the bond in question. This tells us the percentage change of the price of the bond for a one percent change in the interest rates. Elasticity is best thought of as an approximation to the modified duration. Note that there is a question included in this section which is intended for discussion on the main bulletin board.

Question 3 focuses on mortgage valuation. Mortgages differ from corporate or government bonds in how principle is repaid. Recall that bonds pay interest only until maturity, at which time the last interest payment and all of the principle is paid. With mortgages, the same payment is made each period. Part of the payment is for interest. The rest pays off some of the principle. This question involves a fixed rate mortgage. A discussion question for the main bulletin board is included which involves the option that the homeowner has to refinance his mortgage if rates go down.

The final question deals with forward rates. A forward rate is a rate that we can lock in at a later date. The term structure of interest rates has information about forward rates imbedded in it. For example, if the one year rate is 3% and the two year rate is 5%, then we can calculate the forward one year rate for one year away. This forward rate is just the rate that will guarantee us a 5% return if we invest for a year in the one year bond, and then for another year at the forward rate. So, the following relationship will hold:

1.05^{2} = 1.03 (1+_{1}*f*_{2})

The subscripts mean "a one period forward rate at period 2." Solving for the forward rate, we see that the one year rate will have to be about 7.04% in a year in order to guarantee a 5% return per year for a two year investment. In this case, we have an upward sloping term structure. A lot of people have studied the information in the term structure regarding future interest rates. We talked about the information regarding future real growth in the economy. The expectations theory of the term structure which you will study in Macroeconomics says that the forward rate contains some information about future interest rates.

A discussion question is included.

**a)** You have a $1000 par 6% coupon (nominal rate) US Treasury
bond with 7 years remaining in its life. Coupons are paid semiannually
and the next coupon payment is exactly six months away. The market interest
rate is 6.4% (nominal rate with semiannual compounding). What is the current
price of this bond?

**b)** What is the effective annual rate which corresponds to a nominal
interest rate of 6.4% with semi-annual compounding?

**c)** Price a zero coupon bond with a face amount of $1000 maturing
in 7 years. Assume that the nominal interest rate is 6.4% and interest
is compounded semiannually.

**d)** Assume now that interest rates have instantaneously increased
by 1% to 7.4%. What are the bonds in parts **(a)** and **(c) **now
worth?

**2. Duration**

Consider a newly issued $100 par bond with a 10 year maturity and a 7.5% coupon. The first coupon payment is exactly six months away. Coupons are paid semi-annually. The nominal interest rate is 8.5% with semi-annual compounding. Note that the bond price would be equal to par if the nominal market interest rate was 7.5%.

**a)** Calculate the Macaulay duration of this bond.

**b)** Calculate the modified duration of this bond.

**c) **Manually calculate the elasticity of this bond (Hint: Consider
a move in rates from 8.5% to 7.5% and from 8.5% to 9.5% and then average
the percentage changes).

**d)** Compare the actual bond price change in percentage terms as
rates move from 8.5% to 6.5% to the percent price change predicted by your
modified duration calculation.

**Main Bulletin Board Discussion Question:**

In question 2d, the predicted price change misses the actual price change. Why?

**3. Mortgage Amortization**

**a)** You want to take out a 15 year fixed rate mortgage on a $200,000
house. Current mortgage rates are 9.5% (nominal rate, compounded monthly).
Calculate your **monthly **payment on this mortgage.

**b)** Using a spreadsheet, prepare an amortization schedule for
this mortgage showing principal outstanding and the portion of each month’s
payment towards interest and principal.

**c)** After exactly one year (i.e., immediately after the 12th payment),
you use your $75,000 annual bonus to reduce the principal outstanding on
your mortgage. You continue making the same monthly payment that was calculated
in part **(a)**. When will the loan be fully paid off?

**Main Bulletin Board Discussion Question:**

If rates fall, homeowners often renegotiate their mortgage. What kind of option is this? Who writes the option? Who buys it?

**4. Forward Interest Rates**

**a)** Suppose that the current one year interest rate is 5.3% per
annum. Also assume that the 1 year forward interest rate is 5.5% (_{1}*f*_{2}).
This forward rate means that you are able to commit to investing $*x*
one year from now and be certain of receiving $*x*(1+_{1}*f*_{2})
two years from now. How much money will you have in two years if you invest
$100 in the current one year rate (the spot rate) and commit to investing
the proceeds of the one year investment at the one year forward rate? Assume
that interest is calculated annually.

**b)** Assume that investing $100 at the current 2 year interest
rate will leave you with the same amount of money that you calculated in
part (a) at the end of two years. What is the current 2 year rate?

**Main Bulletin Board Discussion Question:**

In the above example, are interest rates expected to rise or fall? Why?