Open the Excel file called spot rates which contains 1-year and 2-year spot rates from June 1976 to July 2020.

a. Calculate the mean and standard deviation for each column of data. What is the relation between the average yield and maturity? What is the relation between the volatility of yields and maturity? Is this the pattern you would expect to observe? Why or why not? Repeat this analysis on two subsamples by dividing your data into two periods, June 76 to April 1994 in one sample, and May 1994 to July 2020 in the other. What differences emerge when you look at average rates and volatility of rates in these two different time periods?

b. The real purpose of working with these numbers is to test the expectations hypothesis of the term structure of interest rates. Remember that this hypothesis says that forward rates are unbiased predictors of future spot rates.

Clearly if you are testing whether forward rates predict future spot rates, you have to begin by taking the data you have and calculate the forward rates implied by the spot rates at any moment in time. You have monthly yields on 2 securities, and from those there just one forward rate you can compute. For example, in June 1976 the 1-year and 2-year spot rates are 6.52% and 7.06% respectively. From those two numbers we know the 1-year forward rate, 1 year in the future is 7.60%. Under the expectations hypothesis, this is the rate wed expect to see on a 1-year Tbill in June 1977.

With this forward rate in hand, I would compare the 1-year spot rate in June 1997 to the 1-year forward rate calculated based on data in June 1976 and calculate a forecast error like this:

Forward rate spot rate = forecast error

7.60% 5.80% = 1.80%

So for this particular period, the forward rate predicted a higher spot rate than what actually prevailed in the market a year later. To conduct a general test of the expectations hypothesis, I want to repeat this comparison every June throughout my sample period. My next step then would be to use data in June 1977 to calculate the one-year forward rate one year ahead, then compare that forward rate to the spot rate in June 1978 to get the forecast error for that period. Just repeat this process until youve gone all the way through the data.

Note in my example above I was careful to not use overlapping data. The reason for this is related to the concept of statistical independence. It is better if each observation in your data is independent of the others. My analysis above uses data in June 1976 to make predictions about June 1977. If I repeated that analysis using data from July 1976 to make predictions about July 1977 (in other words, if I used overlapping data), the results would not be independent because those two observations share 11 monthsthey overlap. So in analyzing the expectations hypothesis, we are going to ignore the data except what we have for June each year.

Once your calculations are complete, you could ask questions like:

(1) Is the one-year forward rate an unbiased predictor of the one-year spot rate one year into the future (remember, an unbiased predictor is one that has a zero forecast error on average)? If not, what is the size and direction of the bias, and what does that imply about different theories of the yield curve?

(2) How much volatility is there in the forecast error (remember, a biased forecast can be quite useful if the bias is predictable)?

(3) If you run a regression in which the Y (dependent) variable is the one-year spot rate at a particular time and the X (independent) variable is the one-year forward rate calculated one year earlier, what do the regression results tell you?

Y = a + bX

Spot rate at time (t) = a + b(forward rate one year earlier)

What value do you expect for a and b if the expectations hypothesis is true? What values for a and b do you expect if the forecast error is biased (remember, the bias could be positive or negative)? What does the regression R-square value tell you?

(4) Investors in bonds are particularly interested in changes in interest rates. Go back to my original example in which I calculated the one-year forward rate starting in June 1976. That rate was 7.60%. Given that the one-year spot rate in June 1976 was 6.52%, the forward rate was predicting an interest rate increase of 1.08%. One year later, the one-year spot rate was 5.80%, so the actual change in interest rates was a decrease of 0.72% (one-year rates went down from 6.52% to 5.80%). Now think about running your regression again, except this time let Y be the actual change in the one-year spot rate (so -0.72% in my example) and let X be the predicted change (from forward rates) from one-year ago (+1.08% in my example).

Y = a + bX

Actual change in spot rate = a + b(forecasted change in spot rates)

Again, if the expectations hypothesis is true what values do you expect for a and b in this regression? How does the R-square value compare here to the previous regression?

Economists often refer to the term random walk. There are two main implications when some economic variable like stock prices, exchange rates, or interest rates follows a random walk. The first is that the future is basically unpredictable...random. The second is that if you must guess what the future value of some quantity will be, the most reasonable forecast is the current value. For example, if the US dollar follows a random walk relative to other currencies, then my best guess for the exchange rate between dollars and euros one year in the future is just todays exchange rate. When evaluating the quality of your interest rate forecasts based upon forward rates, try to assess whether forward rates provide a forecasting tool that is superior to simple random walk estimates. Remember, superior in forecasting has several dimensions including unbiasedness and reliability. Think about different ways you could measure reliability.

Problem 2

The table below lists the price, maturity (in years), and coupon rate (stated in annual terms) for several bonds, all of which pay interest semiannually.

Bond Price	Maturity	Coupon Rt.
1027.07	1	5.000%
1056.60	1	8.000%
1066.42	1	9.000%
1089.37	2	7.250%
1106.33	2	8.125%
1162.12	2	11.000%
1196.56	3	10.000%
1182.24	3	9.500%
1310.99	3	14.000%
1114.07	4	6.750%
1282.19	4	11.250%
1392.60	5	13.000%
1039.62	5	5.250%
1161.92	6	7.875%
1062.30	6	6.000%
1448.20	7	12.500%
1079.52	7	6.375%

Use regression analysis to estimate the term structure. Your deliverable is a series of spot interest rates at 6-month intervals going out 7 years. State each spot rate as an annual rate. Also calculate the YTM for each bond and compare that to the spot rate for the year in which the bond matures.

Let me give you some significant hints about how to set up and interpret the regression. In regression analysis, we usually write equations like this:

Y = a + B1X1 + B2X2 + B3X3 + ....

Here, X1, X2, and X3 are variables that influence Y, and B1, and B2, and B3 are coefficients that measure the extent to which the variables affect Y. In the present context, Y is the price of a bond. We want to know what determines bond prices. That's easy, it's just cash flows and discount rates, and it is the discount rates, the spot rates, which we are after here.

Set up a spreadsheet where each bond is a separate row. The first column is the bond's price. The second column is the cash payment that the bond makes in 6 months. The 3rd column is the cash payment that the bond makes in 12 months. The 4th column is the cash payment that the bond makes in 18 months, and so on. You keep adding data to columns until you have gone far enough out to capture all 14 payments that a 7-year bond would make. Note, after a bond matures, you enter a zero for all payment dates after maturity. For example, the first row of the spreadsheet would look like this:

1027.07 25 1025 0 0 0 0 0 0 0 0 0 0 0 0

Now, use Data Analysis in Excel or some other method to run a multiple linear regression. Price is the Y variable and cash flows in each of the 14 payment periods are the X variables. That is, there are 14 explanatory variables here, and you will get 14 Beta coefficients. Each coefficient is telling you how much an additional dollar of cash flow at a given time period adds to the price of the bond. Notice, I did not say that there should be an intercept term in this regression. There should not be (i.e., no little a as in the regression equation above). You have to instruct Excel or whatever statistical software you are using to force the intercept through zero. Why are we doing this?

Once you have the beta coefficients, youll have to use them to back out what the spot rates are at each point on the yield curve. Youll have to derive the spot rates algebraically. By convention, let's all assume the usual semi-annual compounding for bonds where the denominator of the bond pricing equation would look like (1 + r/2)^t where t is the number of periods (1, 2, 3, etc.) and not the number of years (0.5, 1, 1.5, etc.).

Problem 3

Suppose the expectations hypothesis is true and you have the following series of spot rates.

1-year 1.25% 2-year 1.75% 3-year 2.25% 4-year 2.75%

a. Calculate the price of a bond that has a $1,000 par value, pays a $45 coupon once a year, and matures in four years. You can assume the first coupon is one year away.

b. Assuming the expectations hypothesis is true, what is the expected price of this bond in two years just after it makes its second coupon payment?

c. If you buy the bond today and sell it in one year (just after getting the first coupon payment), what is your expected rate of return.

d. You buy the bond today and hold it to maturity. After you receive a coupon payment, you reinvest it in a zero-coupon bond maturing at the same time (4 years from now) as the coupon-paying bond. How much total cash do you expect to have after 4 years? Given that estimate, what compound annual rate of return do you earn over the 4 years?

e. Go back to the beginning of this problem (use the spot rates given), and assume that investors require a liquidity premium of 0.25% per year to invest in a 2-year bond instead of a bond maturing in year 1. For example, investors want to earn 0.25% more per year if they invest in a two-year bond than they expect to earn if they buy a one-year bond and then roll that over into another one-year bond. In other words, now we are assuming that the expectations hypothesis is NOT true. Does your answer to part (a) change? Why or why not? If you are now asked to predict the bonds price in one year, is your answer different than in part (c)? Note that I am not asking you to derive a price. Just tell me if it would be different or not and explain why. If you think it would be different, would be the price be higher or lower than your answer to part (c )?

f. Considering the information given in part (e), what is the expected interest rate on a one-year bond one year from today?

Problem 4

The current 1-year spot rate is 3%, and ₁f₁ is 3.5%. Investors require a liquidity premium such that the expected annual return on a 2-year bond is 0.35% per year higher than the expected return on a series of two, 1-year bonds. What is the current market price of a 2-year bond that pays a 4% coupon rate (annual payments) and what is the expected price of this bond one year from today? If you buy the bond today, hold it for one year, collect its first coupon payment, then immediately resell it at the expected selling price, what is your expected return?

Problem 5

Calculate the duration, in years, of a bond that matures in 10 years, pays $40 in interest twice a year (e.g., 8% coupon rate), and sells for $973.29. Explain how you can use duration (or modified duration) to estimate the change in the price of a bond when the yield to maturity changes. Estimate how much the price of this bond will change if rates go up one percentage point, and then calculate an exact price assuming a one percentage point jump in the YTM. Why doesnt the duration approach give you an exact answer?

Problem 6

A bond sells at par and pays an annual coupon of 5% (assume annual payments). Calculate the duration of this bond at various maturities including 1, 2, 3, 4, 5, 10, 20, 50, and 100 years. Plot a graph showing how the duration changes as maturity changes. Explain why the graph is shaped the way it is. Be careful on your graph that the units on the x-axis progress

Problem 7

Your firm is obligated to make payments to a supplier according to the following schedule:

Year 1 $1.5 million

Year 2 $1.75 million

Year 3 $10 million

You want to immunize this liability investing in some combination of a 2-year bond with a 4% coupon and a 4-year bond with a 6% coupon. You can assume these bonds pay interest annually and have a par value of $1,000. Use a discount rate / YTM of 6% throughout this problem. How many (in terms of numbers of bonds) of each type of bond do you need to purchase to immunize your liability?