Reference no: EM132396425

Computer assignment – The Term Structure of Interest Rates

Part 1

This assignment asks you to build a spreadsheet template (excel) to derive the term structure of interest rates from market quotes on treasury bills and treasury notes.

On an excel sheet, you are asked to use market prices of US treasury securities to compute the market spot and forward interest rates, as well as the term premiums, the term structure, duration and modified duration of US bonds of varying maturities. It is recommended to read the relevant chapters in the Bodie, Kane and Marcus book.

When submitting the assignment please submit the results (a Word file or Pdf is fine) as well as an excel sheet in which you performed the calculations. Note: in addition to the numerical answers you should write 1-3 lines interpreting yours results, giving some intuition.

1. Constructing the term structure

Assume that today is Friday September 27, 2019. From the Wall Street Journal look at the quoted bid and ask discounts on treasury bills for September 27, 2019 (attached). Use the November 29, 2019 treasury bill to obtain the first interest rate in the constructed term-structure.
Explanation: Remember that the interest rates are the YTMs on zero-coupon government bonds. T-bills are indeed zero coupon bonds, so the YTM on a T-bill is the interest rate from now to the time the T-bill matures.

Because today the date is September 27, 2019, the first maturity in the term structure you are constructing is 63 days (30-27=3 days in September plus 31 days in October plus 29 days in November). The relevant column is the ASKED YIELD because this is the YTM for investors who buy the bonds today.

This means that the 63-days interest rate in the market was 1.818% on September 27, 2019. This is the first point on the yield curve in the assignment. Remember that the yield curve relates the time to maturity to YTM on zero-coupon government bonds. This means that the first point in the graph is as follows: The X- axis is 63 days, and the Y-axis is 1.818%. Notice that all yields in the WSJ are in annual terms.

Comment about fees: the fact that we are buying the securities at the ask price means we are paying a transaction fee. Note that the ask yield is always lower than the bid yield. This means that the ask price (the price we pay if we actually purchase the bond) is higher than the price we can sell it for since prices are inversely related to yields.

Now we want to find the next point on the yield curve. Let us progress in increments of half a year. So since the first point is the end of November 2019, the next point will be the end of May 2020, the following point will be the end of November 2020, then the next will be the end of May 2021 and so on.

Next look at the TREASURY NOTES&BONDS data. Consider the bond that matures on May 31, 2020 (we will call this bond “the May 2020” bond). When there are several bonds maturing on a given date, always pick the one with the highest coupon rate (so the May 2020 bond is the one with coupon rate of 1.375%).

Notice that bonds in the US pay coupons every 6 months. This bond matures on May 31 2020 so it will pay the last coupon and the par on that date and it will also pay a coupon 6 months before maturity, i.e. on November 30, 2019.

Compute the total price today (September 27, 2019) of the May 2020 bond. This price includes the quoted asked price plus accrued interest (important: you must add the accrued interest to the price on the internet. See the book, chapter 14, on how to calculate the accrued interest). Notice that this note has 246 days to maturity (30- 27+31+30+31+31+28+31+30+31).

Comment: the yields in WSJ are in annual terms. They are not exactly yields  to maturity as we learned in class. Explanation: consider for example a bond with 2 years to maturity, coupon rate 4%, and par 1000 and market price today is 950. Recall that this is a US bond, which pays coupons every six months. The WSJ finds first the semi-annual yield as the solution for y in the following equation:

950 = 20/(1+y) + 20/(1+y)^2 + 20/(1+y)^3 + 1020/(1+y)^4

And then the WSJ multiplies y by 2 and defines 2*y as the annual yield. So this is actually not exactly the YTM in annual terms. Instead it is the semi-annual YTM multiplied by 2. 2*y is called the Bond Equivalent Yield (BEY). What you are asked to do in this assignment is to build the term structure by relating the times to maturity to the BEY of the bonds, so you need to first find the YTM’s in semi-annual terms and then multiply them by 2.

Deduct from the May 2020 bond’s total price the value of the last coupon payment before maturity (meaning the end of November 2019 coupon), using the November 29 T-bill rate as the discount rate. Explanation: we are deducting the present value of the first coupon for the bond. The reason to do this is that we want to calculate the value of only the last coupon plus the par value. That value is essentially the value of a “zero-coupon bond” that matures on May 31, 2020 (where the payoff at maturity date is the par plus last coupon).

Once having that value, we can calculate  the YTM on that “zero coupon bond” and this YTM will be the interest rate from today to 246 from now. Note: the May 2020 bond will actually pay its next coupon on November 30, 2019 so we should have used the interest rate from today to 64 (not 63 days from now). However, given that we do not have T-bill maturing on November 30, we will use the one maturing on November 29 as a proxy.

What will be the discount rate to use for the first coupon? It is 1.818% in annual terms. First, let’s bring this to semi-annual terms. The semi-annual YTM is 1.818%/2 = 0.909%. Next, we need to bring this into 64-days term. To do this we calculate 1.00909^(64/180). This is the discount rate for the first coupon. Note: the coupon will be paid on November 30 2019 but we are using the interest rate from today to November 29 as the discount rate as an approximation.

After deducting from the May 2020 bond the PV of its first coupon, we are now in a position to find the second term on the term structure. How to do this? You have the price of the last coupon + par, and you know how much will be the last coupon + par. So you can calculate the rate of return from today to 246 days from now. You will then need to convert this rate of return to effective semi-annual terms and multiply by two to get the interest rate for 246 days expressed as BEY. In doing so you will find the second term on the yield curve.

After that, proceed in a similar manner with the bond maturing on November 30 2020 (always use the last bond in the month even if the maturity is not exactly the last day of the month) – use the November 29 2019 T-bill rate and the discount rate you computed from the May 2020 bond to strip from the November 2020 bond price the value of its coupons to be paid in the end of November 2019 and on May 31 2020.

Calculate the discount rate on a zero bond that pays a sum equal to the par and last coupon payment of the November 2020 note. Proceed further by increasing maturities in 6 months increments to construct pure discount bonds (zero bonds) with maturities of up to 5 years (so the last interest rate you should calculate is the one from today to November 30, 2024).

Note: on the yield curve each term should be converted to effective semi-annual terms and multiplied by 2 (just as the WSJ compute it).

2. Compute the term premium for each maturity. The term premium is the difference between the yield on a given bond and the yield on the bill with the shortest maturity (the November 29 2019 bill).

3. For each pair of securities with adjacent maturities, compute the annualized forward interest rate implied by the term structure you have calculated in (1). Again, as in question 1, convert the rates into effective semi-annual and multiply by 2.

4. Graph the term structure of interest rates, i.e. the yield curve and the forward interest rates (the rates should be on the vertical axis and time should be on the horizontal axis).

5. According to the expectations hypothesis, what is the 6-months interest rate that investors expect will be in the market on May 31, 2020? Please explain. Can we say something on that expected interest rate if we believe in the liquidity preference theory? Under which theory that expected interest rate is higher?

6. According to the expectations hypothesis, what is the expected realized yield on the November 2020 note?

Part 2

1. If the yield curve is flat, then under the expectations hypothesis the market expects that in the future the interest rate will rise. True or false? Please explain.

2. The following prices of zero coupon bonds with par 1000 are given:

Time to maturity	Price
1	922.40
2	842.52
3	805.37

a. Consider a 3-year coupon bond with coupon rate 8.5% paid once a year and par 1000. What is the YTM on this bond?

b. What will be the HPR on the bond if we buy it now and in one year the yield curve will be flat at 7.5%?

3. The following term structure of interest rates is given

Time to maturity	YTM
1	6.0%
2	6.1%
3	6.2%
4	6.3%

a. If you think that the term structure of interest rates a tear from now will be the same as it is now, will a 4-year zero coupon bond bought now and sold after one year give an HPR which is lower or higher than the HPR on a 1-year zero coupon bond bought toda?

b. Is your expectation for the term structure next year consistent with the expectations hypothesis?

4. Assume that a pension fund has an obligation to pay 10 million USD in five years time. Assume that the YTM on a zero coupon government bond with 5 years to maturity is 6.5% (in annual terms). The fund has as assets only cash at the amount which is equal to the present value of its obligation. Assume that the fund is interested in immunizing itself by investing in bonds with zero coupon, par 1000 and 9 years to maturity and by investing in bonds with 2 years to maturity, par 1000 and coupon rate 8% paid once a year and YTM 2.463%. Assume that the net worth of the fund is zero (i.e. market value of liabilities equals market value of assets).

a. Construct a portfolio from the 9 years and 2 years bonds which will protect the fund from interest rate fluctuations. What are the weights of the 9 year bonds and the weights of the 2 year bonds in that portfolio and how much money should be invested in each bond?

b. If immediately after the fund bought the 2-year coupon bonds and the 9 year zero coupon bonds the yield curve changed and became flat at 7.2%, what will be the fund’s portfolio new duration? Will the weights of the 2 types of bonds change if the fund would like to stay immunized?

5. Suppose that there was news that unemployment is higher than expected. What will happen to bond yields and prices? Answer in no more than 3 lines.

6. Suppose that the term structure today is given as follows:

Time to maturity	YTM
1	1.1%
2	1.5%
3	2.0%
4	2.8%

Suppose your investment horizon is 4 years, and you are considering buying a coupon bond with par 1000, coupon rate 3%, paying once a year.

a. Assume you believe in the expectations hypothesis. What is your expected realized yield?

b. If you believe in the liquidity preference theory, will your expected realized yield be lower, equal or higher than the one in (a)? Why? Answer in no more than 3 lines.

7. Suppose that the price of a T-bill maturing one year from now is 950. Suppose also that the two-years interest rate is 6%, the three-years interest rate is 6.5% and the four-years interest rate is 6.8%.

You are considering two investment strategies for four years. The first is by investing in T-bills and each year rolling over your money for one year until four years from now. The second is buying a 4-years to maturity, coupon rate 4%, paid once a year, and par 1000, and reinvesting the coupon for one year at a time.
 
a. What are the expected realized yields on each of the two strategies if you believe in the expectations hypothesis.

b. What will be the realized yields if it turns out that in 8-months from now the yield curve will become flat at 3% and will stay that way until four years from now?

8. Assume that your investment horizon is 5 years. You consider buying a 10- years to maturity coupon bond with coupon rate 7% paid once a year and par 1000. Assume that the yield curve today is flat at 6%.

a. Would you prefer a scenario where the yield curve will rise to being flat at 6.5% in 2-months from now and stay that way for the next five years and a half or a scenario where the yield curve will fall to become flat at 5.5% 2-months from now and stay at that level for the next five years and a half?