Getting Started: Defines terms that are important to know for building a yield curve.

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1 Word Capital Open Source Asset Management 408 West 14 th Street, 2F New York, NY US Yield Curve Tutorial Jake Roth Caroline Davidson Tools Needed 1 Microsoft Excel 2 Access to a Bloomberg Terminal Contents 1 Getting Started: Defines terms that are important to know for building a yield curve. 2 Components and Formulas: Explains several concepts used for relating components of the yield curve. 3 Step-by-Step Guide: Gives instructions for extracting and using data to create a curve.

2 Getting Started The yield refers to the return on an investment. A yield curve plots the interest rates for investments that mature at different points in time. It shows what rate should be used for lending at various maturities. We will use deposit rates, Eurodollars, and swap rates to build our US yield curve. The US Dollar yield curve is the most relevant application to all global markets participants. Vivamus porta est sed est. Deposit rates are the interest rates that banks charge for lending and pay for borrowing at different maturities. Eurodollar futures contracts are deposits of US dollars in foreign banks that have a 3-month maturity with contracts starting every month. The price of the futures contracts is related to the interest rate offered. We will use contracts starting in March, June, September, and December. The deposits are made outside of the United States, so they are not subject to Federal Reserve regulation. The swap rate is the rate at which the fixed-rate payment in an interest rate swap is traded. An interest rate swap is the exchange of a fixed-rate payment for a floating-rate payment on a standardized notional. The yield to maturity refers to the rate of return on a bond at the end of its maturity. It is essentially an average of the yields for the different time periods on payments throughout its maturity. We take the yield to maturity into account when using swap rates to compute the yield for the 2-year to 30- year portion of the curve. Simple interest rates are most commonly used when dealing with maturities lasting less than a year. We use simple interest rates when dealing with the deposit rates and Eurodollars. Compounding interest rates are rates where interest is added to the notional so that, from that moment on, the interest that has been added also earns interest. We deal with compounding interest rates when we start using swap rates to determine points on the yield curve. The swap rates we deal with are compounded annually. Day-count conventions are used to calculate the interest rates between two dates. In building a yield curve, we often need to compute the interest rate for periods between payments or another period, which requires using a reference period for which we already know the rate. We use the equation:

3 Components and Formulas Today 3 months Deposit rates Deposit rates are short term, so we use them to determine the portion of the curve from the current date to the start of the first available Eurodollar contract. We use the spot-next deposit rate (which we will refer to as the 1-day rate), 1-week, 2-week, 3-week, 1-month, 2-month, and 3- month rates. In order to get the rates for days in between these tenors, we use linear interpolation and the theory of similar triangles. Here, we know i upper, i lower, upper, and lower: Using the endpoints of the triangle, we can create a new, smaller triangle within the larger one. We know that the ratios between these right triangles are equivalent, so we can solve for ix using the equation below and plugging in the information we know. This formula can be expressed in a way to solve for any interest rate between two known data points:

4 3 months 2 years Eurodollars Eurodollar contracts last for 90 days (3 months), and we use the contracts that start in March, June, September, and December (or in market terminology: EDH, EDM, EDU, EDZ). They extend to 10 years; however, we use them only to calculate the 3-month to 2-year portion of the yield curve. Because the rates represent only a 90 day time period that starts and ends at different points in time, it is important to account for the interest rate relative to the portion of time before that rate is put into effect. Using the rate for the previous term (i 1) and the rate for the 3-month contract (i 2), we can solve for itotal using the equation: To solve for the interest rate for the total period of time, we manipulate the equation: To calculate the interest rate for the next Eurodollar contract, i 1 becomes the previous i total. This equation always breaks the total time period into two segments: the time period that we have already dealt with and the newest piece of time that we are adding on. Bloomberg gives data in percent form, but the equation requires that the interest rates be in decimal form. Yet we want to use interest rates in percentage form for the yield curve, so it is important to make the conversions when necessary, i.e. multiplying or dividing by 100 when appropriate.

5 2 years 30 years Swap rates We use swap rates to determine the 2-year to 30-year portion of the yield curve. However, the data for swap rates gives the yield to maturity, and we must calculate the yield for the final portion of the time to maturity while incorporating interest rates for the previous periods of time. Now we know i 1 and i total, and we are solving for i 2. We use the equation: To calculate the yield, solve for in in the equation above and manipulate it to become: Again, coupons (C) are given in percent form, so the appropriate conversion to decimal form is required. When calculating rates, we assume the bonds are priced at par. A bond priced at par means that its future cash flows, when brought to present day terms, are equal to the given price.

6 Step-by-Step Guide Data 1. Use the Bloomberg Terminal to extract data for deposit rates up to 3 months using the BLP() function. You can find data on the deposit rates by typing BBC 33, <GO>. 2. Extract data for Eurodollars up to 2 years. Use the formula 100- BLP() to determine the rate. Also, extract data for the expiration date of each contract by using the BLP() function with LAST_TRADEABLE_DT as the second input. Data on Eurodollar rates can be found by typing EDSF, <GO>. 3. Extract data for swap rates up to 30 years. Use the BLP() formula to retrieve rate data. Swap rates can be found by typing IRSB 42, <GO>. Insert rows between 10, 15, 20, 25, and 30 years to account for all years in between. Dates and Day Counts 1. Create a column for the date that starts with the current date using Excel s TODAY() function. The column should continue to increment to the next day until 30 years from now. 2. Create a column next to it for the day count. It should start at 0 and increment up 1 for each subsequent day. 3. For both columns, each cell should add 1 to the value in the cell before it. Fill this in for the first three increments, then select the second and third cells and drag down for all days. V-Lookup Table 1. Create a V-Lookup table of 4 columns. This table will be used to get the upper and lower bounds needed for linear interpolation. The first two columns are maturity lengths and the corresponding rate. The next two columns should be identical to the first two, but shifted up one row. (See V- Lookup Table Excel Example) 2. To begin with, only fill this table with deposit rates (1-90 day rates). Rates on longer maturities will be calculated and added to the table later. For example, the overnight information on the left side should be in line with the 1- week information on the right side. Linear Interpolation 1. Use the VLOOKUP() function to get the upper and lower bounds for any given day and create a table using the Date and Day Count columns. (See Linear Interpolation Excel Example). 2. Set your VLOOKUP() formula so that the final input is TRUE, which tells the function to find the first closest match. It will always find a hit on the lower bound of a number. 3. Use the VLOOKUP() with the first input being the specific day. Get the rates of the lower and upper bounds in columns 2 and 4. Use this same logic to obtain the upper and lower bound day values. 4. Use linear interpolation in the yield column. Deposit Rates 1. Use the information in the v- lookup table and linear interpolation to solve for the interest rate at each day. 2. Drag the formula and appropriate information all the way through the table for each day. The first column is the lower bound day, and the following columns are the lower bound day s rate, the upper bound day, the upper bound day s rate, and then the formula calculating ix in the final column. After the deposit rate is calculated the other yield rates will be added to this table as well. 3. Be sure to lock the V-Lookup table in place before dragging down. Otherwise, the VLOOKUP() will search through a table filled with incorrect information. Eurodollars 1. When the first Eurodollar contract goes into effect, the formula must be changed: 2. Use the rate for the previous time period (i1) and the rate for the next 3-month Eurodollar contract (i2) to solve for itotal in the equation above. Both pieces of information must be used to compute the rate for the total period of time each time a new Eurodollar futures contract comes into effect. 3. Plug this data into the V-Lookup table. Using the same logic from the deposit rates, get the lower and upper bounds and use linear interpolation to calculate the rates for the days in between. Swap Rates 1. Use the equation below and input appropriate data to calculate interest rates for 3 years to 30 years. 2. This formula may become overwhelming with the denominator growing to 31 terms, so implement a table to simplify the formula.

3. Factor out the coupon value from the discount factor terms in the denominator. Now the formula will look this: ` Linear Interpolation Excel Example 4.

Calculate them all at once, and then create a table with columns that grow by one term each year.

Now condense the formula by using the SUM() function to add all the discount factors in the appropriate year.

7 3. Factor out the coupon value from the discount factor terms in the denominator. Now the formula will look this: ` Linear Interpolation Excel Example 4. The discount factors for each year will be consistent over time (values in each rows are the same). Calculate them all at once, and then create a table with columns that grow by one term each year. This will result in a table similar to the Discount Factors Table Excel Example. 5. Now condense the formula by using the SUM() function to add all the discount factors in the appropriate year. Then, distribute the coupon value to return to the original equation. 6. After solving for the interest rates, plug the values into the V- Lookup table. 7. Using linear interpolation, calculate the rates for the days in between. Discount Factors Table Excel Example Graph 1. Select the Date and Yield columns of the data. 2. Insert a Line Graph. 3. Title and label axes appropriately. V-Lookup Table Excel Example

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Computing interest and composition of functions:

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