This chapter defines interest rate formats used within Oracle Asset Liability Management (ALM) and Funds Transfer Pricing, describes which format is used in a particular process and what type or types of conversion are used; and outlines the algorithms used for such conversions.
Interest rates come in a variety of formats. Within Oracle Asset Liability Management (ALM) and Funds Transfer Pricing (FTP), interest rates are used for multiple purposes, with each rate requiring a specific format. The system must apply conversion formulas to translate the interest rates from their starting format into the format proper for its use in any given process.
Topics:
The following characteristics define an interest rate code:
· accrual basis
· compound basis
· rate format
The accrual basis can be:
· 30/360
· 30/365
· 30/Actual
· Actual/Actual
· Actual/365
· Actual/360
· Business/252
NOTE:
See Stigum, M., and Robinson Money Market and Bond Calculations, Irwin, 1981, for definition.
The compound basis can be:
· daily
· monthly
· quarterly
· semi-annual
· annual
· simple
The rate format can be
· zero-coupon yield
· yield-to-maturity
· discount factor
The discount factor is used only internally and cannot be specified as an input rate format in the Rate Management Interest Rate setup. For bonds issued at par with payment frequency equal to the compound basis, the yield-to-maturity at origination is equal to the coupon. There are several definitions of yield-to-maturity. The unconventional true yield definition of Stigum is not used. Instead, the Street convention is preferred.
|
Symbol |
Name |
Notes |
|---|---|---|
|
AI(Ti) |
Accrued interest for the i-th security. |
|
|
C(Ti) |
Coupon value of the i-th security. |
This is the true $value of the cash flow (not annualized) |
|
m |
Compounding frequency (per year). |
Possible values are: 12 – monthly 2 – semi-annual 1- annual 0- Simple. |
|
ni |
Number of full compounding periods from As of Date up to term Ti |
|
|
P(T) |
The discount factor with term Ti |
Value of a zero-coupon bond with term Ti and par=$1. |
|
r |
The total number of securities. |
|
|
Ti |
The term in Act/Act years of the i-th security. |
Sorted in ascending order. |
|
L(i,k) |
Time in Act/Act years of the start of the k-th (k=0…ni) compounding period for security i. |
|
|
wi |
Residual, that is, the number of compounding periods between the current date and the next compounding date for i-th security. |
|
|
yTM(Ti) |
Yield-to-maturity of the i-th security |
|
|
yzc(Ti) |
The zero-coupon yield of the i-th security |
|
NOTE:
See Fabozzi, F. The Handbook of Fixed Income Securities. McGraw Hill, 1977.
The yield curve is composed of r par bonds with different terms. Par value is equal to $1.
The zero-coupon yield is the vector of r values yzc(Ti) that solve the following set of r equations:
Equation1
Description of Rate Conversion Equation 1 follows
If compounding is simple, or
Equation 2
Description of Rate Conversion Equation 2 follows
otherwise
The yield-to-maturity for the i-th security is the value yTM(Ti) that solves the equation.
Equation 3
Description of Rate Conversion Equation 3 follows
If compounding is simple, or
Equation 4
Description of Rate Conversion Equation 4 follows
otherwise.
Scenario-based forecast rates are the interest rates used for repricing events, rate setting of new business, and market-rate observations for prepayment assumptions and new business assumptions. The Forecast Rates assumption rule stores the definition of forecasted rate scenarios.
Within the Forecast Rates assumption rule, the user creates deterministic scenario(s) for each IRC, defining the forecasted rates in each modeling bucket (for an active time bucket rule) for each IRC and scenario. Most methods use the as-of-date rates, stored in history tables, as the basis for the rates forecast. The following methods are available:
· Flat
· Structured Change
· Direct Input
· Change from Base
· Implied Forward
· Yield Curve Twist
The flat method assumes rates in effect on the as of date remain constant for the entire term structure over the modeling horizon.
The user inputs a series of rate changes that can vary over the yield curve terms and the modeling horizon. At runtime and display time, the rate changes are added to the as-of-date rates to create a future scenario. No conversion is applied before the rate is passed to the Cash Flow Engine.
The Direct Input method requires the user to input (or upload) of the forecasted rate scenarios. At runtime and display time, the inputted rates will be pulled directly.
Before applying the rates to a cash flow record, the rates must be converted to annual compounded rates. The accrual basis must be converted to either Actual/Actual or 30/360.
No conversion is applied before the rate is passed to the Cash Flow Engine.
For a change from the base, the user defines a shock to the rate forecast from another scenario, the base scenario. At runtime and display time, the system must first calculate the base scenario rate and then apply the shock to the rate forecast. No conversion is applied before the rate is passed to the Cash Flow Engine.
The implied forward calculation assumes that the rate is a zero-coupon yield. If the IRC rate format is yield-to-maturity the following process occurs:
· Translate input rates from yield-to-maturity to zero-coupon yield.
· Apply implied forward calculations on zero-coupon yields.
· Translate results of implied forward calculation from zero-coupon yields back into yield-to-maturity.
For IRCs that are already in zero-coupon yield format, the implied forward calculations can be applied directly to the historical rates, and no conversion between formats is required.
The user inputs the tenors representing the Short Point, Anchor Point, and Long Point. For each of these tenor points, the user additionally adds the required shock amounts for each tenor. At runtime and display time, the rate changes are added to the as-of-date rates to create a future scenario. No conversion is applied before the rate is passed to the Cash Flow Engine.
With the Forecast Rates assumption rule, exchange rates can be forecast as a function of the interest rate scenarios. Each currency has a reference yield curve that drives the parity exchange rate relationships between this currency and other currencies. The two methods that require interest rate forecasts are forward and parity.
If the reference IRC is in a yield-to-maturity format, the system converts the rate to a zero-coupon yield format before calculation of the exchange rate.
In scenario-based processes, users select, per product and currency, an IRC as their valuation curve. Market value sensitivity processes use shocks to the as-of-date rates to drive the market value changes. Dynamic market value calculations use the forecasted curve as of the dynamic valuation date. The set of valuation curves can be determined from the Discount Methods assumption rule. The system converts all rates to zero-coupon format before processing.
Monte Carlo processes require annually compounded Act/Act zero-coupon yields as the input. If the input IRC format is anything other than that, a conversion process is applied.
Stochastic rates output from the Monte Carlo engine is also annually compounded Act/Act zero-coupon yields.
In the Stochastic Rate Index assumption rule users select an IRC as their valuation curve. For each currency, formulas are applied to the yields forecasted from the valuation curve for the Monte Carlo process. The formula for each additional IRC must exist in the Stochastic Rate Index assumption rule. The formulas are applied during processing when one of the additional IRCs is required for pricing new business or repricing existing business.
There are both cash flow and non-cash flow transfer pricing methods in Oracle FTP.
The non-cash flow transfer pricing methods use historical IRC data pulled directly from the historical rate tables. For these methods, the format of the IRC used as the transfer pricing yield curve is assumed to be a yield-to-maturity.
There are four cash flow methods:
· Weighted Average Term
· Average Life
· Duration
· Zero-Coupon
Weighted Average Term: Weighted Average Term calculates the cash flows over the funding period, treating the next repricing date as the maturity date. The cash flows are discounted by the current net rate. The discounted cash flow at each payment/maturity is used as the weighting factor for the rate from the transfer pricing yield curve. The term from the origination to the cash flow date is used as the term for the lookup to the transfer pricing yield curve.
For this method, the transfer pricing yield curve is assumed to be in the proper rate format. No adjustment will be made to the current net rate or the transfer pricing yield curve.
Average Life: The average life method calculates the average life by taking the cash flows over the funding period and calculating average life for the series of cash flows. The computed average life of the cash flows is used as the term for the lookup to the transfer pricing yield curve.
For this method, the transfer pricing yield curve is assumed to be in the proper format. No adjustment will be made to the current net rate.
Duration: The duration method calculates the duration by taking the cash flows over the funding period and calculating the duration for the series of cash flows. The current net rate is used as the discount rate. The duration of the cash flows is used as the term for the lookup to the transfer
The duration of the cash flows is used as the term for the lookup to the transfer pricing yield curve. For this method, the transfer pricing yield curve is assumed to be in the proper format. No adjustment will be made to the current net rate.
Zero Discount Factors: The zero discount factor method must calculate discount factors for the transfer pricing yield curve. If the transfer pricing yield curve is stored as yield-to-maturity rates, the rates must first be translated into zero-coupon yields so that the discount factor can be calculated from them. If the transfer pricing yield curve is already in zero-coupon yield format, then discount factors can be calculated directly from the rates.
Conversion of accrual basis or compounding basis is straightforward. Rate format conversions (zero-coupon and yield-to-maturity) are more difficult.
Based on Equation 9-3 and Equation 9-4, the system first calculates the values
c(Ti)
AI(Ti)
Then the system bootstraps the yield curve using the BFGS algorithm to solve Equation 9-1 nd Equation 9-2.
NOTE:
See, for instance, Press W., Flannery, Teukolsky, and Vetterling. Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press, Second Edition, 1992.
Based on equations Equation 9-1 and Equation 9-2, the system first calculates the values
c(Ti)
AI(Ti)
Then, the system solves Equation 9-1 and Equation 9-2 using the Newton-Raphson algorithm.