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Yield Curve CreationLocate this document in the navigation structure

Use

This function enables you to create yield curves. A yield curve is defined by the reference interest rates assigned to it. The system uses the market data that is available for the individual reference interest rates to create the yield curve, that is, the system calculates discount factors and zero interest rates at the grid points defined on the basis of the reference interest rates.

Features

Features

The system calculates discount factors using the bootstrapping procedure. This procedure ensures that the discount factors are calculated on the basis of the specified market data without arbitrages. The system carries out the following steps:

  1. Find market data

    The system searches the interest rates for a specific date for all of the references interest rates assigned to the yield curves. The Read Back Directly procedure is used when a maximum age is specified:

    • Determine the read date using the condition Date Before or Same as Specified Date for the reference interest rates assigned to the yield curve. The system uses the specified date to search for the most recent date in the past on which market data is available; in other words, the system continues to set the read date back one day, starting from the specified date, until it finds an interest rate for at least one assigned reference interest rate.

    • If the read date found exceeds the maximum age (read date before the specified date minus the maximum age), the system responds in the same way as it would if no market date were available. In this case, no yield curve is created.

    • If the read date is not too old, the system reads the available interest rates from the market data table for this date and for all the reference interest rates assigned to the yield curve. The system uses the interest rates found for the read date to create the yield curve for the specified date. This yield curve is then valid for the specified date.

      Note

      You create the permitted maximum age (in days), in Customizing for Cross-Application Components under Start of the navigation path Market Data Next navigation step Interest Next navigation step Edit Reference Interest Rates and Yield Curves End of the navigation path.

  2. Calculate payment amounts and times for the assigned reference interest rates

    The system uses the interest rate conditions of the reference interest rates to calculate the payments and payment times of the individual reference interest rates for which market data was found:

    • Calculate the start of the first interest period:

      This date is calculated from the validity date plus the number of working days for interest fixing. The system uses the calendar specified for the reference interest rate to determine the working days and public holidays. If you have not specified a calendar, the system ignores the number of working days for interest fixing, that is, the start date of the first interest period is, in this case, the same as the validity date. The start of the first interest period is the date on which the capital is paid in for the interest rate instrument underlying the reference interest rate.

    • Calculate payment dates:

      The system uses the entries made for the reference interest rate in the Term/Time Unit and Payment Frequency fields to calculate the payment dates, starting from the first interest period.

      Example

      Term: 3 years, payment frequency: annual, start of the first interest period: 2011–05–17

      The payment dates 2012–05–17, 2013–05–17, and 2014–05–17 are created.

    • Depending on the Customizing setting you have chosen for the Shift Value Date to Working Day indicator for the reference interest rate, the system shifts the payment dates determined in the previous step to working dates in accordance with the specified rule. The shifted payment dates are used in the next step.

    • Calculate the quotients from the number of days divided by the number of days in the year for each individual interest period of the reference interest rate in line with the interest calculation method specified in Customizing for the reference interest rate. The payment date of the previous interest period (or start of the first interest period if the current interest period is the first interest period) is always the start of an interest period and the payment date of the current interest period is the end of an interest period.

    • Calculate the individual interest payments:

      Multiply the interest rate found by the quotients determined in the previous step from the number of days divided by the number of days in the year. A zero interest rate with a term of more than one year, however, results in the interest portion C of the repayment, as illustrated in the figure below:



      where R: zero interest rate, q: quotient from the number of days divided by the number of days in the year.

    • The system then sets the date of the capital repayment to the payment date of the last interest period.

  3. Sort the market data

    The system sorts the interest rates found in ascending order by term and then calculates the discount factors recursively in exactly this order.

  4. Calculate the discount factors

    Together with the quoted interest rate, each reference interest rate implicitly provides a well-defined payment flow of n interest payments Ci (calculated in the Calculate payment amounts and times for the assigned reference interest rates step), including a repayment of 100. The quoted interest rates are quoted at par, that is, the net present value (NPV) of the cash flow of all future payments is the same as its remaining debt of 100. To calculate the net present value, the system multiplies each individual payment (interest and capital repayment) by a discount factor di. This results in the following NPV equation (d0 is usually equal to 1, d0 is not equal to 1 when the start date of the first interest period is not the same as the validity date), as illustrated in the figure below:



    .

    The system now makes use of the fact that the yield curve has been created by time Tn-1 and, therefore, that the discount factors d0 to dn-1 of payment times T0 to Tn-1 have already been established. Established here means that these discount factors have either already been calculated or that they can be obtained by interpolation. To calculate dn, the above equation is resolved as illustrated in the figure below:



    .

    This enables you to obtain discount factors for all the terms specified by the reference interest rates. This procedure is called a ‘bootstrapping procedure’.

  5. Calculate discount factors for zero interest rates with a term of more than one year

    For zero interest rates with a term of more than one year, the system calculates the discount factors using a different procedure to the one above, as illustrated in the figure below:



    .

    For zero interest rates with a term of more than one year, a compounding frequency of one year is, therefore, assumed.

  6. Calculate the discount factor d0

    If the payment time T0 for the discount factor d0 is after all the payment times (grid points) that have already been created in the yield curve, the system assumes that the continuous compounding zero interest rates between the last available grid point and T0, and between T0 and the first interest payment date are the same. The validity date of the yield curve for which the discount factor is always equal to 1 also implicitly applies as the grid point.

    If, however, a grid point exists with a longer term, the system interpolates d0.

    Example

    The reference interest rate with the shortest term in the yield curve has a term of one month, the payment frequency is monthly, and the start of the interest period is two working days after the interest rate has been fixed (after the validity date of the interest rate).

    In this case, the system assumes that the continuous compounding zero interest rates between the validity date of the yield curve/interest rate and the start of the interest period and between the start and the end of the interest period are the same. On the basis of this assumption, the system can calculate d0.

  7. Fill swap gaps

    There are two different methods for filling swap gaps from which you can choose:

    1. Fill swap gaps by interpolation of par interest rates (Customizing: Linear Interpolation of Continuously Compounded Zero Rates calculation method)

      The system does not always recognize all discount factors d0 to dn-1. If a yield curve is assigned reference interest rates for 1, 2, 3, 4, 5, 7, and 10-year terms, for example, the grid points (the swap gaps) for 6, 8, and 9 years are missing, which enables the system to use the equation specified above. For example, the discount factor d6 is missing for the determination of d7. The system cannot yet perform an extrapolation to determine d6, since the yield curve contains gaps. For this reason, the system has to fill these gaps in another way, namely by linear interpolation, that is, the system interpolates the par rates of the reference interest rates with the terms 5 and 7 years in a linear manner and to the exact day in order to calculate the interest rate for 6 years. This is illustrated in the figure below:



      In this example, it is assumed that an interest payment is made once a year for all interest rates.

      For each term Ti of the reference interest rates concerned, the actual number of days divided by 365 has to be applied. This interpolation of par interest rates is only performed during the creation of the yield curve in order to fill the swap gaps. If, however, the yield curve does not contain gaps, the system performs interpolation in accordance with the procedure described in the Interpolation section. If the interest rate conditions of two successive reference interest rates are different (if the interest calculation methods R5 and R7 are different, for example), the system uses the interest calculation method R7 at time T5 to calculate an interest rate to be quoted at par, and applies this interest rate in the formula above. For more information about calculating interest rates, see Calculation of Interest Rates from the Yield Curve Created. To enable the system to apply the formula above, the interest rate conditions of the interest rates concerned must be identical (with the exception of the term).

      Once the system has filled the gaps, it calculates the discount factors that are still missing (see Calculate the Discount Factors). The yield curve is created once all discount factors have been calculated up to the grid point with the longest term.

    2. Fill swap gaps by interpolation of zero interest rates (Customizing: Linear Interpolation of Zero Interest Rates (also for swap gaps)

      Here, the system does not calculate additional grid points as in the Linear Interpolation of Continuously Compounded Zero Rates method. Instead it linearly interpolates the zero interest rate of the previously calculated grind point and the unknown zero interest rate for the grid point that is to be entered, and uses these to calculate the net present values of interest payments to the swap gaps. Since the comparison that occurs cannot be solved algebraically, the system uses Newton's iteration. Then no more additional grid points are displayed in the yield curve for the swap gaps. In the following example, the yield curve is created for a term of five years. The next grid point to be entered is that for seven years. The system calculates net present value PV for the grid point to be entered using the equation depicted in the figure below:



      Discount factors d6 and d7 are unknown. The figures below illustrate the discount factors d6 and d7:





      These discount factors have the correct value if the equation illustrated in the figure below is satisfied:



      The system uses Newton's iteration method to solve the equation as follows:

      • For the NPV equation above the system represents zero interest rate Z6 for six years as a linear interpolation of zero interest rate Z5 (already known) and Z7 (unknown), as the figure below illustrates:



        Terms Ti are - both here and above - the actual number of days divided by 365. This means that zero interest rate Z7 is now the only unknown variable needed for the calculation of discount factors d6and d7 for the above NPV equation.

      • The system runs Newton's iteration method with the above NPV equation for unknown zero interest rate Z7. It selects a starting value for zero interest rate Z7 and changes it using iteration steps until the value found for Z7 satisfies the equation illustrated in the figure below:



      • Then the system calculates discount factor d7 and inserts it together with Z 7 as the grid point for seven years in the yield curve. It is not necessary to insert a grid point for six years here because zero interest rate Z6 for 6 years is calculated using linear interpolation between Z5 and Z7.