Sensitivity Key Figures 

Use

Sensitivity key figures describe the interest rate risk of securities. They are usually calculated for fixed-rate securities.

If you do not consider the creditworthiness of the borrower or market efficiency, then fixed-rate securities are subject only to the risk of a change in market interest rates during their term. This interest rate risk comprises reinvestment risk and rate risk. All repayment flows (principal repayments, interest, and compound interest) for an investment that occur before the investor’s planning horizon, are exposed to reinvestment risk. For example, if market interest rates are falling, the coupon payments can only be reinvested at lower interest rates. If the maturity date of the investment exceeds the planning horizon, then there is an additional rate risk because the rate is governed during the term of the transaction by the market interest rate.

Market interest rate changes have opposite effects on both these yield components of an investment: increasing interest rates give rise to rate losses on the one hand but increasing reinvestment yields on the other. Sensitivity key figures enable you to quantify and manage such interest rate change risks and opportunities more easily.

Features

The system enables you to calculate the sensitivity key figures Macaulay duration, Fisher-Weil duration, convexity, and the basis point sensitivity.

Macaulay Duration

Macaulay duration describes the term that an investment needs to have so that the counteracting effects of rate change and reinvestment yield offset each other exactly. Macaulay duration can be interpreted as the average commitment period in years of the invested capital. Using this point in time as the planning horizon, the investor is then immune from any interest rate changes.

The Macaulay duration of a portfolio is the mean value of the Macaulay durations calculated for the individual transactions and weighted by their NPVs.

The system calculates the Macaulay duration of individual transactions as follows:

·       Macaulay Duration of a Fixed-Rate Transaction

The Macaulay duration of a fixed-rate transaction is the quotient of the total of the cash flows weighted at the points in time of the payments, and the NPV of the transaction on the horizon.

where CF_i are the cash flows of the transaction at time point t_i, t_horizon is the horizon, and d_i are the discount factors at time point t_i on horizon date t_horizon. The system calculates the Macaulay duration in years.

·       Macaulay Duration of Variable-Rate Transactions

For a variable transaction, the system takes market data from the time period between the horizon and the maturity date of the last coupon whose interest rate fixing data is before the horizon.

where t_maturity is the maturity date of the coupon, and t_horizon is the horizon. The system calculates the Macaulay duration in years. If the variable side is after the horizon, then the market data is zero.

·       Macaulay Duration of Swaps

The Macaulay duration of a swap is the weighted total of the Macaulay duration for the variable and fixed-rate parts, with the summands being weighted by the proportion of the respective swap side in the NPV of the total swap.

where NPV_var is the NPV of the variable side of the swap; NPV_fix is the NPV of the fixed side of the swap on the horizon date; NPV_swap is the NPV of the swap (or NPV_var+NPV_fix); CF_i are the cash flows of the transaction at time point t_i; t_horizon is the horizon date; d_i is the discount factor from time point t_i to the horizon date t_horizon,; and t_maturity is the maturity date of the last variable-rate coupon whose interest rate fixing date is before the horizon. The system calculates the Macaulay duration in years.

 

Note that the calculation of the Macaulay duration for variable-rate transactions and swaps gives useful results only if the yield curves for adding accrued interest, and discounting, are identical. However, in the SAP system you can use various yield curves for forward rates, and discounting. There is currently no suitable theoretical model for Macaulay duration for this case.

 

Macaulay duration is not calculated for options, since this is not relevant from a business perspective.

 

The calculations used to determine the Macaulay duration for variable-rate transactions and swaps come from Albrecht/Stephan (1993) [Single-factor immunizing duration of an interest rate swap, Proceedings of the 4th AFIR International Colloquium, Orlando, 1994, Vol. 2, pp. 757-780].

 

Fisher-Weil Duration

Fisher-Weil duration describes the elasticity of the NPV to interest rate changes.

The SAP system calculates the Fisher-Weil duration as the difference quotient of the NPVs by shifting e the market rates upon which they are based:

where NPV is the net present value of the transaction, and NPV(+e) and NPV(-e) is the NPV of the transaction after the market rates have been shifted upwards or downwards. A basis point is used as the value of the shift (e = 0.01% = 0.0001). The system calculates the Fisher-Weil duration in years.

The Fisher-Weil duration shows by how many basis points the value of an interest rate instrument changes when the level of the interest rate changes by one basis point. Therefore the change in the NPV can be understood as follows:

The Fisher-Weil duration of a portfolio is the mean value of the Fisher-Weil durations calculated for the individual transactions and weighted by their NPVs.

 

Convexity

The convexity is the sensitivity of the NPV to changes in the interest rate described by the curvature of the price curve.

The system calculates the convexity as the difference quotients by shifting e the underlying market interest rates.

where NPV is the net present value of the transaction, and NPV(+e) and NPV(-e) is the NPV of the transaction after the market rates have been shifted upwards or downwards. A basis point is used as the value of the shift (e = 0.01% = 0.0001). The factor 10^-2 is the standardization.

The convexity of a portfolio is the mean value of the convexities calculated for the individual transactions and weighted by their NPVs.

 

Sensitivity per Basis Point (Price Value of Basis Point, PVBP)

The basis point value describes the change in the market value in the event of an increase in market interest rates for all terms, each by one basis point (0.01%). It specifies absolute changes.

 

Activities

You have two options for calculating sensitivity key figures. Either use the functions in the results database or, in the SAP Easy Access screen choose Accounting ® Bank Applications ® SEM Banking ® Market Risk Analysis ® Information System ® Sensitivity Key Figures.

 

Additional Notes

Diverging Values for Fisher-Weil Duration and Convexity

The values calculated by the system for Fisher-Weil duration and convexity can become very large if the NPV of the transaction is very small. This is often the case for derivatives. For swaps, negative and positive cash flows can almost net each other off.

For this reason, it is possible that the values calculated for Fisher-Weil duration and convexity for particular transactions cannot be interpreted properly. Since the system aggregated the values weighted by their NPVs, the fact that the values diverge is not problem if the relevant transactions are in the same base portfolio as loans, for example.

Dependency of the Continuous Compounding Indicator

How the key figures for Macaulay duration and Fisher-Weil duration are calculated depends on whether you set the Continuous Compounding indicator for the yield curve type used. If you set the indicator, then the system uses constant zero rates to interpolate the interest rates, and not annual rates. It also applies the act/365 interest calculation method, and regardless of other settings, it uses the linear interpolation method.

In these conditions, the values calculated for Macaulay duration and Fisher-Weil duration should be the same, provided one or multiple fixed cash flows are used for the calculation, and exact formulas, formulated by derivation, are taken into account. If the cash flows are linked to a reference rate, then the values for the two types of duration are the same if the same yield curve is used for the calculation of forward rates and for discounting. Since the system uses difference quotients to calculate an approximation of the sensitivity key figures, the values calculated for Macaulay duration and Fisher-Weil duration are usually the same for only one deterministic cash flow. For more complex products, and for variable-rate products, the results are usually different.

Example of a Single, Deterministic Cash Flow

For single, deterministic cash flows, the values calculated for sensitivity key figures depend only on the term of the cash flow and the interest rate that is valid. The following formulas contain the NPV, interest rate, and the residual term of the cash flow:

where t_maturity is the maturity date of the coupon, and t_horizon is the horizon. The following example of a single, deterministic cash flow is based on the assumption that exact formulas have been used for the sensitivity key figures:

·       If the Continuous Compounding indicator is not set, and the term of the cash flow is up to one year, then the system uses the linear interest calculation method:

 

·       If the Continuous Compounding indicator is not set, and the term of the cash flow is one year or more, then the system uses the exponential interest calculation method:

·       If the Continuous Compounding indicator is set, then the system uses the constant interest calculation method, regardless of the term of the cash flow: