rate2disc (Financial Derivatives Toolbox)

Discounting factors from interest rates

Syntax

Usage 1: Interval points are input as times in periodic units.

Disc = rate2disc(Compounding, Rates, EndTimes, StartTimes)

Usage 2: ValuationDate is passed and interval points are input as dates.

[Disc, EndTimes, StartTimes] = rate2disc(Compounding, Rates, 
EndDates, StartDates, ValuationDate)

Arguments

Compounding
Scalar value representing the rate at which the input zero rates were compounded when annualized. This argument determines the formula for the discount factors:
Compounding = 1, 2, 3, 4, 6, 12
Disc = (1 + Z/F)^(-T), where F is the compounding frequency, Z is the zero rate, and T is the time in periodic units, e.g. T = F is one year.
Compounding = 365
Disc = (1 + Z/F)^(-T), where F is the number of days in the basis year and T is a number of days elapsed computed by basis.
Compounding = -1
Disc = exp(-T*Z), where T is time in years.

Rates
Number of points (NPOINTS) by number of curves (NCURVES) matrix of rates in decimal form. For example, 5% is 0.05 in Rates. Rates are the yields over investment intervals from StartTimes, when the cash flow is valued, to EndTimes, when the cash flow is received.

EndTimes
NPOINTS-by-1 vector or scalar of times in periodic units ending the interval to discount over.

StartTimes
(Optional) NPOINTS-by-1 vector or scalar of times in periodic units starting the interval to discount over. Default = 0.

EndDates
NPOINTS-by-1 vector or scalar of serial maturity dates ending the interval to discount over.

StartDates
(Optional) NPOINTS-by-1 vector or scalar of serial dates starting the interval to discount over.
Default = ValuationDate.

ValuationDate
Scalar value in serial date number form representing the observation date of the investment horizons entered in StartDates and EndDates. Required in Usage 2. Omitted or passed as an empty matrix to invoke Usage 1.

`Compounding`	Scalar value representing the rate at which the input zero rates were compounded when annualized. This argument determines the formula for the discount factors: `Compounding` = `1`, `2`, `3`, `4`, `6`, `12` `Disc = (1 + Z/F)^(-T)`, where `F` is the compounding frequency, `Z` is the zero rate, and `T` is the time in periodic units, e.g. `T = F` is one year. `Compounding` = `365` `Disc = (1 + Z/F)^(-T)`, where `F` is the number of days in the basis year and `T` is a number of days elapsed computed by basis. `Compounding` = `-1` `Disc = exp(-T*Z)`, where `T` is time in years.
`Rates`	Number of points `(NPOINTS)` by number of curves (`NCURVES`) matrix of rates in decimal form. For example, 5% is 0.05 in `Rates`. `Rates` are the yields over investment intervals from `StartTimes`, when the cash flow is valued, to `EndTimes`, when the cash flow is received.
`EndTimes`	`NPOINTS`-by-`1` vector or scalar of times in periodic units ending the interval to discount over.
`StartTimes`	(Optional) `NPOINTS`-by-`1` vector or scalar of times in periodic units starting the interval to discount over. Default = `0`.
`EndDates`	`NPOINTS`-by-`1` vector or scalar of serial maturity dates ending the interval to discount over.
`StartDates`	(Optional) `NPOINTS`-by-`1` vector or scalar of serial dates starting the interval to discount over. Default = ValuationDate.
`ValuationDate`	Scalar value in serial date number form representing the observation date of the investment horizons entered in `StartDates` and `EndDates`. Required in Usage 2. Omitted or passed as an empty matrix to invoke Usage 1.

Description

Disc = rate2disc(Compounding, Rates, EndTimes, StartTimes) and [Disc, EndTimes, StartTimes] = rate2disc(Compounding, Rates, EndDates, StartDates, ValuationDate) convert interest rates to cash flow discounting factors. rate2disc computes the discounts over a series of NPOINTS time intervals given the annualized yield over those intervals. NCURVES different rate curves can be translated at once if they have the same time structure. The time intervals can represent a zero curve or a forward curve.

Disc is an NPOINTS-by-NCURVES column vector of discount factors in decimal form representing the value at time StartTime of a unit cash flow received at time EndTime.

StartTimes is an NPOINTS-by-1 column vector of times starting the interval to discount over, measured in periodic units.

EndTimes is an NPOINTS-by-1 column vector of times ending the interval to discount over, measured in periodic units.

If Compounding = 365 (daily), StartTimes and EndTimes are measured in days. The arguments otherwise contain values, T, computed from SIA semiannual time factors, Tsemi, by the formula T = Tsemi/2*F, where F is the compounding frequency.

The investment intervals can be specified either with input times (Usage 1) or with input dates (Usage 2). Entering ValuationDate invokes the date interpretation; omitting ValuationDate invokes the default time interpretations.

Examples

Example 1.

Compute discounts from a zero curve at six months, 12 months, and 24 months. The time to the cash flows is 1, 2, and 4. We are computing the present value (at time 0) of the cash flows.

Compounding = 2;
Rates = [0.05; 0.06; 0.065];
EndTimes   = [1; 2; 4];
Disc = rate2disc(Compounding, Rates, EndTimes)

Disc =
    0.9756
    0.9426
    0.8799

Example 2.

Compute discounts from a zero curve at six months, 12 months, and 24 months. Use dates to specify the ending time horizon.

Compounding = 2;
Rates = [0.05; 0.06; 0.065];
EndDates = ['10/15/97'; '04/15/98'; '04/15/99'];
ValuationDate = '4/15/97'; 
Disc = rate2disc(Compounding, Rates, EndDates, [], ValuationDate)

Disc =
    0.9756
    0.9426
    0.8799

Example 3.

Compute discounts from the one-year forward rates beginning now, in six months, and in 12 months. Use monthly compounding. The times to the cash flows are 12, 18, 24, and the forward times are 0, 6, 12.

Compounding = 12;
Rates = [0.05; 0.04; 0.06];
EndTimes = [12; 18; 24];
StartTimes = [0; 6; 12];
Disc = rate2disc(Compounding, Rates, EndTimes, StartTimes)
Disc =
    0.9513
    0.9609
    0.9419

See Also

disc2rate, ratetimes

optbndbyhjm ratetimes