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The *interest-rate term structure* represents
the evolution of interest rates through time. In MATLAB^{®}, the
interest-rate environment is encapsulated in a structure called `RateSpec`

(*rate specification*). This structure
holds all information required to completely identify the evolution
of interest rates. Several functions included in Financial Instruments Toolbox™ software
are dedicated to the creating and managing of the `RateSpec`

structure.
Many others take this structure as an input argument representing
the evolution of interest rates.

Before looking further at the `RateSpec`

structure,
examine three functions that provide key functionality for working
with interest rates: `disc2rate`

,
its opposite, `rate2disc`

, and `ratetimes`

. The first two functions map
between discount factors and interest rates. The third function, `ratetimes`

, calculates
the effect of term changes on the interest rates.

*Discount
factors* are coefficients commonly used to find the current
value of future cash flows. As such, there is a direct mapping between
the rate applicable to a period of time, and the corresponding discount
factor. The function `disc2rate`

converts
discount factors for a given term (period) into interest rates. The
function `rate2disc`

does
the opposite; it converts interest rates applicable to a given term
(period) into the corresponding discount factors.

As an example, consider these annualized zero-coupon bond rates.

From | To | Rate |
---|---|---|

15 Feb 2000 | 15 Aug 2000 | 0.05 |

15 Feb 2000 | 15 Feb 2001 | 0.056 |

15 Feb 2000 | 15 Aug 2001 | 0.06 |

15 Feb 2000 | 15 Feb 2002 | 0.065 |

15 Feb 2000 | 15 Aug 2002 | 0.075 |

To calculate the discount factors corresponding to these interest
rates, call `rate2disc`

using the syntax

Disc = rate2disc(Compounding, Rates, EndDates, StartDates, ValuationDate)

where:

`Compounding`

represents the frequency at which the zero rates are compounded when annualized. For this example, assume this value to be 2.`Rates`

is a vector of annualized percentage rates representing the interest rate applicable to each time interval.`EndDates`

is a vector of dates representing the end of each interest-rate term (period).`StartDates`

is a vector of dates representing the beginning of each interest-rate term.`ValuationDate`

is the date of observation for which the discount factors are calculated. In this particular example, use February 15, 2000 as the beginning date for all interest-rate terms.

Next, set the variables in MATLAB.

StartDates = ['15-Feb-2000']; EndDates = ['15-Aug-2000'; '15-Feb-2001'; '15-Aug-2001';... '15-Feb-2002'; '15-Aug-2002']; Compounding = 2; ValuationDate = ['15-Feb-2000']; Rates = [0.05; 0.056; 0.06; 0.065; 0.075];

Finally, compute the discount factors.

Disc = rate2disc(Compounding, Rates, EndDates, StartDates,... ValuationDate)

Disc = 0.9756 0.9463 0.9151 0.8799 0.8319

By adding a fourth column to the rates table (see Calculating Discount Factors from Rates) to include the corresponding discounts, you can see the evolution of the discount factors.

From | To | Rate | Discount |
---|---|---|---|

15 Feb 2000 | 15 Aug 2000 | 0.05 | 0.9756 |

15 Feb 2000 | 15 Feb 2001 | 0.056 | 0.9463 |

15 Feb 2000 | 15 Aug 2001 | 0.06 | 0.9151 |

15 Feb 2000 | 15 Feb 2002 | 0.065 | 0.8799 |

15 Feb 2000 | 15 Aug 2002 | 0.075 | 0.8319 |

The function `rate2disc`

optionally returns
two additional output arguments: `EndTimes`

and `StartTimes`

.
These vectors of time factors represent the start dates and end dates
in discount periodic units. The scale of these units is determined
by the value of the input variable `Compounding`

.

To examine the time factor outputs, find the corresponding values in the previous example.

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

Arrange the two vectors into a single array for easier visualization.

Times = [StartTimes, EndTimes]

Times = 0 1 0 2 0 3 0 4 0 5

Because the valuation date is equal to the start date for all
periods, the `StartTimes`

vector is composed of 0s.
Also, since the value of `Compounding`

is 2, the
rates are compounded semiannually, which sets the units of periodic
discount to six months. The vector `EndDates`

is
composed of dates separated by intervals of six months from the valuation
date. This explains why the `EndTimes`

vector is
a progression of integers from 1 to 5.

The function `rate2disc`

also
accommodates an alternative syntax that uses periodic discount units
instead of dates. Since the relationship between discount factors
and interest rates is based on time periods and not on absolute dates,
this form of `rate2disc`

allows
you to work directly with time periods. In this mode, the valuation
date corresponds to 0, and the vectors `StartTimes`

and `EndTimes`

are
used as input arguments instead of their date equivalents, `StartDates`

and `EndDates`

.
This syntax for `rate2disc`

is:

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

Using as input the `StartTimes`

and `EndTimes`

vectors
computed previously, you should obtain the previous results for the
discount factors.

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

Disc = 0.9756 0.9463 0.9151 0.8799 0.8319

The function `disc2rate`

is
the complement to `rate2disc`

.
It finds the rates applicable to a set of compounding periods, given
the discount factor in those periods. The syntax for calling this
function is:

Rates = disc2rate(Compounding, Disc, EndDates, StartDates, ValuationDate)

Each argument to this function has the same meaning as in `rate2disc`

. Use the results found in the
previous example to return the rate values you started with.

Rates = disc2rate(Compounding, Disc, EndDates, StartDates,... ValuationDate)

Rates = 0.0500 0.0560 0.0600 0.0650 0.0750

As in the case of `rate2disc`

, `disc2rate`

optionally returns `StartTimes`

and `EndTimes`

vectors
representing the start and end times measured in discount periodic
units. Again, working with the same values as before, you should obtain
the same numbers.

[Rates, EndTimes, StartTimes] = disc2rate(Compounding, Disc,... EndDates, StartDates, ValuationDate);

Arrange the results in a matrix convenient to display.

Result = [StartTimes, EndTimes, Rates]

Result = 0 1.0000 0.0500 0 2.0000 0.0560 0 3.0000 0.0600 0 4.0000 0.0650 0 5.0000 0.0750

As with `rate2disc`

, the relationship between
rates and discount factors is determined by time periods and not by
absolute dates. So, the alternate syntax for `disc2rate`

uses
time vectors instead of dates, and it assumes that the valuation date
corresponds to time = 0. The time-based calling syntax is:

`Rates = disc2rate(Compounding, Disc, EndTimes, StartTimes);`

Using this syntax, you again obtain the original values for the interest rates.

Rates = disc2rate(Compounding, Disc, EndTimes, StartTimes)

Rates = 0.0500 0.0560 0.0600 0.0650 0.0750

`bdtprice`

| `bdtsens`

| `bdttimespec`

| `bdttree`

| `bdtvolspec`

| `bkprice`

| `bksens`

| `bktimespec`

| `bktree`

| `bkvolspec`

| `bondbybdt`

| `bondbybk`

| `bondbyhjm`

| `bondbyhw`

| `bondbyzero`

| `capbybdt`

| `capbybk`

| `capbyblk`

| `capbyhjm`

| `capbyhw`

| `cfbybdt`

| `cfbybk`

| `cfbyhjm`

| `cfbyhw`

| `cfbyzero`

| `fixedbybdt`

| `fixedbybk`

| `fixedbyhjm`

| `fixedbyhw`

| `fixedbyzero`

| `floatbybdt`

| `floatbybk`

| `floatbyhjm`

| `floatbyhw`

| `floatbyzero`

| `floatdiscmargin`

| `floatmargin`

| `floorbybdt`

| `floorbybk`

| `floorbyblk`

| `floorbyhjm`

| `floorbyhw`

| `hjmprice`

| `hjmsens`

| `hjmtimespec`

| `hjmtree`

| `hjmvolspec`

| `hwcalbycap`

| `hwcalbyfloor`

| `hwprice`

| `hwsens`

| `hwtimespec`

| `hwtree`

| `hwvolspec`

| `instbond`

| `instcap`

| `instcf`

| `instfixed`

| `instfloat`

| `instfloor`

| `instoptbnd`

| `instoptembnd`

| `instoptemfloat`

| `instoptfloat`

| `instrangefloat`

| `instswap`

| `instswaption`

| `intenvprice`

| `intenvsens`

| `intenvset`

| `mmktbybdt`

| `mmktbyhjm`

| `oasbybdt`

| `oasbybk`

| `oasbyhjm`

| `oasbyhw`

| `optbndbybdt`

| `optbndbybk`

| `optbndbyhjm`

| `optbndbyhw`

| `optembndbybdt`

| `optembndbybk`

| `optembndbyhjm`

| `optembndbyhw`

| `optemfloatbybdt`

| `optemfloatbybk`

| `optemfloatbyhjm`

| `optemfloatbyhw`

| `optfloatbybdt`

| `optfloatbybk`

| `optfloatbyhjm`

| `optfloatbyhw`

| `rangefloatbybdt`

| `rangefloatbybk`

| `rangefloatbyhjm`

| `rangefloatbyhw`

| `swapbybdt`

| `swapbybk`

| `swapbyhjm`

| `swapbyhw`

| `swapbyzero`

| `swaptionbybdt`

| `swaptionbybk`

| `swaptionbyblk`

| `swaptionbyhjm`

| `swaptionbyhw`

- Modeling the Interest-Rate Term Structure
- Pricing Using Interest-Rate Term Structure
- Pricing Using Interest-Rate Term Structure
- Pricing Using Interest-Rate Tree Models
- Graphical Representation of Trees