computing of interest rates

Mini-case 1 (provide all the answers in an Excel file)

Mini-case 1 consists of a series of problems and a case study. The main purpose of this mini-case is to solidify your understanding of interest rates.

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Problem 1 (20 points) – understanding how to compute duration

Consider two bonds: bond XY and bond ZW. Bond XY has a face value of \$1,000 and 10 years to maturity and has just been issued at par. It bears the current market interest rate of 7% (i.e. this is the yield to maturity for this bond). Bond ZW was issued 5 years ago when interest rates were much higher. Bond ZW has face value of \$1,000 and pays a 13% coupon rate. When issued, this bond had a 15-year, so today its remaining maturity is 10 years. Both bonds make annual coupon payments.

a) (5 points) What is the price of Bond ZW, given that market interest rates are 7%?
b) (15 points) Compute the duration for both bonds (use Excel).

Problem 2 (35 points) – understanding the determinants of duration

In this exercise, you are going to analyze first the relationship between interest rates and bond prices, and then the effect of time to maturity, interest rates and coupon rates on duration.

a) (5 points) First, consider a 10 year bond with a coupon rate of 7% and annual coupon payments. Draw a graph showing the relationship between the price and the interest on this bond. The price should be on the y-axis and the interest rate on the x-axis. To compute the various prices, consider interest rates between 2% and 12% (use 0.5% increments). So your x-axis should go from 2%, then 2.5% … until 11.5% and then 12%.
Is the relationship linear (i.e. is the slope constant)? Start at 7%. If interest rates go up or down by 0.5% is the price changing by the same amount? What type of relationship do we observe between prices and interest rates (liner, concave, convex or something else)?

b) (5 points) Now consider the same bond with 10 year maturity, a face value or \$1,000, a coupon rate of 7% (coupon is paid annually) and assume that the yield to maturity on the bond is 7%. Compute the duration of this bond.

c) (5 points) Next, we are going to analyze the effect of time to maturity on the duration of the bond. Compute the duration of a bond with a face value of \$1,000, a coupon rate of 7% (coupon is paid annually) and a yield to maturity of 7% for maturities of 2 to 18 years in 1-year increments (so here we are going to vary the time to maturity and see how duration changes if N=2, 3 … etc.). What happens to duration as maturity increases?

d) (5 points) Next, we are going to analyze the effect of the yield to maturity on the duration of the bond. Compute the duration of a bond with a face value of \$1,000, a coupon rate of 7% (coupon is paid annually) and a maturity of 10 years as the interest rate (or yield to maturity) on the bond changes from 2% to 12% (consider increments of 1% – so you need to compute the duration for various yields to maturity 2%, 3%, …, 12%) . What happens to duration as the interest rate increases?

e) (5 points) Finally, we are going to analyze the effect of the coupon payment on the duration of the bond. Compute the duration of a bond with a face value of \$1,000, a maturity of 10 years and a yield to maturity of 7%. Compute the duration for coupon rates ranging from 2% to 12% (in increments of 1%). What happens to duration as the coupon rate increases?

Problem 3 (5 points) – understating expected value and standard deviation

You own a \$1,000 face value, zero-coupon bond that has 5 years of remaining maturity. You plan on selling the bond in one year and believe that the required yield next year will have the following probability distribution:

Probability Required Yield
0.1 5.50%
0.1 5.75%
0.6 6.00%
0.1 6.25%
0.1 6.50%

a. What is your expected price when you sell the bond?
b. What is the standard deviation?

Case study (40 points) – understanding the term structure of interest rates

https://hbsp.harvard.edu/import/723841

After reading the case study, answer the following questions (you also need to find Estrella’s study from the NY Fed – a link is provided in the case study). Also, you can use other sources as long as you cite them. To find the current yield curve (as well as historical yield curves you can go to https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/Historic-Yield-Data-Visualization.aspx)

1. (15 points) Estrella (NY Fed) is quite certain that the yield curve is a good predictor of future economic activity. From the case, or the link to his FAQs, answer the following questions:
a. How successful is the yield curve at predicting recessions?
b. What matters most – the level of the term spread, the change in the spread, or the level of short term interest rates?
c. Discuss why a yield curve inversion should lead to a recession.

2. (15 points) Dick Berner (Morgan Stanley) is a bit more skeptical about the predictive power of the yield curve. Does he just not understand Estrella’s overwhelming evidences, or does his skepticism rest on solid reasoning?

3. (10 points) How is the U.S. yield curve currently sloped? What does it affect your forecast of economic activity?

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