## Interest-Rate Term Conversions

Interest-rate evolution is typically represented by a set of interest rates, including the beginning and end of the periods the rates apply to. For zero rates, the start dates are typically at the valuation date, with the rates extending from that valuation date until their respective maturity dates.

### Spot Curve to Forward Curve Conversion

Frequently, given a set of rates including their start and end dates, you may be
interested in finding the rates applicable to different terms (periods). This
problem is addressed by the function `ratetimes`

. This function
interpolates the interest rates given a change in the original terms. The syntax for
calling `ratetimes`

is

[Rates, EndTimes, StartTimes] = ratetimes(Compounding, RefRates, ... RefEndDates, RefStartDates, EndDates, StartDates, ValuationDate);

where:

`Compounding`

represents the frequency at which the zero rates are compounded when annualized.`RefRates`

is a vector of initial interest rates representing the interest rates applicable to the initial time intervals.`RefEndDates`

is a vector of dates representing the end of the interest rate terms (period) applicable to`RefRates`

.`RefStartDates`

is a vector of dates representing the beginning of the interest rate terms applicable to`RefRates`

.`EndDates`

represent the maturity dates for which the interest rates are interpolated.`StartDates`

represent the starting dates for which the interest rates are interpolated.`ValuationDate`

is the date of observation, from which the`StartTimes`

and`EndTimes`

are calculated. This date represents time = 0.

The input arguments to this function can be separated into two groups:

The initial or reference interest rates, including the terms for which they are valid

Terms for which the new interest rates are calculated

As an example, consider the rate table specified in Calculating Discount Factors from 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 |

Assuming that the valuation date is February 15, 2000, these rates represent
zero-coupon bond rates with maturities specified in the second column. Use the
function `ratetimes`

to calculate the
forward rates at the beginning of all periods implied in the table. Assume a
compounding value of 2.

% Reference Rates. RefStartDates = ['15-Feb-2000']; RefEndDates = ['15-Aug-2000'; '15-Feb-2001'; '15-Aug-2001';... '15-Feb-2002'; '15-Aug-2002']; Compounding = 2; ValuationDate = ['15-Feb-2000']; RefRates = [0.05; 0.056; 0.06; 0.065; 0.075]; % New Terms. StartDates = ['15-Feb-2000'; '15-Aug-2000'; '15-Feb-2001';... '15-Aug-2001'; '15-Feb-2002']; EndDates = ['15-Aug-2000'; '15-Feb-2001'; '15-Aug-2001';... '15-Feb-2002'; '15-Aug-2002']; % Find the new rates. Rates = ratetimes(Compounding, RefRates, RefEndDates,... RefStartDates, EndDates, StartDates, ValuationDate)

Rates = 0.0500 0.0620 0.0680 0.0801 0.1155

Place these values in a table like the previous one. Observe the evolution of the forward rates based on the initial zero-coupon rates.

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

15 Feb 2000 | 15 Aug 2000 | 0.0500 |

15 Aug 2000 | 15 Feb 2001 | 0.0620 |

15 Feb 2001 | 15 Aug 2001 | 0.0680 |

15 Aug 2001 | 15 Feb 2002 | 0.0801 |

15 Feb 2002 | 15 Aug 2002 | 0.1155 |

### Alternative Syntax (`ratetimes`

)

The `ratetimes`

function can provide
the additional output arguments `StartTimes`

and
`EndTimes`

, which represent the time factor equivalents to the
`StartDates`

and `EndDates`

vectors. The
`ratetimes`

function uses time
factors for interpolating the rates. These time factors are calculated from the
start and end dates, and the valuation date, which are passed as input arguments.
`ratetimes`

can also use time
factors directly, assuming time = 0 as the valuation date. This alternate syntax
is:

```
[Rates, EndTimes, StartTimes] = ratetimes(Compounding, RefRates,
RefEndTimes, RefStartTimes, EndTimes, StartTimes);
```

Use this alternate version of `ratetimes`

to find the forward
rates again. In this case, you must first find the time factors of the reference
curve. Use `date2time`

for this.

RefEndTimes = date2time(ValuationDate, RefEndDates, Compounding)

RefEndTimes = 1 2 3 4 5

RefStartTimes = date2time(ValuationDate, RefStartDates, ... Compounding)

RefStartTimes = 0

These are the expected values, given semiannual discounts (as denoted by a value
of `2`

in the variable `Compounding`

), end dates
separated by six-month periods, and the valuation date equal to the date marking
beginning of the first period (time factor = `0`

).

Now call `ratetimes`

with the alternate
syntax.

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

Rates = 0.0500 0.0560 0.0600 0.0650 0.0750 EndTimes = 1 2 3 4 5 StartTimes = 0 0 0 0 0

`EndTimes`

and `StartTimes`

have, as expected,
the same values they had as input arguments.

Times = [StartTimes, EndTimes]

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

## See Also

`instbond`

| `instcap`

| `instcf`

| `instfixed`

| `instfloat`

| `instfloor`

| `instoptbnd`

| `instoptembnd`

| `instoptfloat`

| `instoptemfloat`

| `instrangefloat`

| `instswap`

| `instswaption`

| `intenvset`

| `bondbyzero`

| `cfbyzero`

| `fixedbyzero`

| `floatbyzero`

| `intenvprice`

| `intenvsens`

| `swapbyzero`

| `floatmargin`

| `floatdiscmargin`

| `hjmtimespec`

| `hjmtree`

| `hjmvolspec`

| `bondbyhjm`

| `capbyhjm`

| `cfbyhjm`

| `fixedbyhjm`

| `floatbyhjm`

| `floorbyhjm`

| `hjmprice`

| `hjmsens`

| `mmktbyhjm`

| `oasbyhjm`

| `optbndbyhjm`

| `optfloatbyhjm`

| `optembndbyhjm`

| `optemfloatbyhjm`

| `rangefloatbyhjm`

| `swapbyhjm`

| `swaptionbyhjm`

| `bdttimespec`

| `bdttree`

| `bdtvolspec`

| `bdtprice`

| `bdtsens`

| `bondbybdt`

| `capbybdt`

| `cfbybdt`

| `fixedbybdt`

| `floatbybdt`

| `floorbybdt`

| `mmktbybdt`

| `oasbybdt`

| `optbndbybdt`

| `optfloatbybdt`

| `optembndbybdt`

| `optemfloatbybdt`

| `rangefloatbybdt`

| `swapbybdt`

| `swaptionbybdt`

| `hwtimespec`

| `hwtree`

| `hwvolspec`

| `bondbyhw`

| `capbyhw`

| `cfbyhw`

| `fixedbyhw`

| `floatbyhw`

| `floorbyhw`

| `hwcalbycap`

| `hwcalbyfloor`

| `hwprice`

| `hwsens`

| `oasbyhw`

| `optbndbyhw`

| `optfloatbyhw`

| `optembndbyhw`

| `optemfloatbyhw`

| `rangefloatbyhw`

| `swapbyhw`

| `swaptionbyhw`

| `bktimespec`

| `bktree`

| `bkvolspec`

| `bkprice`

| `bksens`

| `bondbybk`

| `capbybk`

| `cfbybk`

| `fixedbybk`

| `floatbybk`

| `floorbybk`

| `oasbybk`

| `optbndbybk`

| `optfloatbybk`

| `optembndbybk`

| `optemfloatbybk`

| `rangefloatbybk`

| `swapbybk`

| `swaptionbybk`

| `capbyblk`

| `floorbyblk`

| `swaptionbyblk`

## Related Examples

- 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