Forward yield: Difference between revisions

Latest revision as of 20:33, 1 July 2022

The fixed interest rate in the market today for an investment or borrowing commitment:

Starting at a fixed future date; and
Ending on a later fixed future date.

The commitment can relate to a physical deposit or borrowing, or - more commonly - a derivative contract to be settled by reference to a notional deposit or borrowing.

For example a forward rate might be quoted for a forward rate agreement for the maturity 2-5 months in the future.

Example

Taking a simpler example, say the forward rate (or forward yield) for the maturity 2-3 periods is 3% per period.

2 is the time from today (into the future) when the investment or borrowing will start.

3 is the time from today when the investment or borrowing will end.

The difference between Time 3 periods and Time 2 periods is the length of the investment or borrowing.

In this case the length of the investment or borrowing is 3 - 2 = 1 period.

Assuming a deposit, 3% is the rate payable for period 3 only - a single period - which is pre-agreed today, 2 periods before the deposit is contracted to change hands.

This means a mutually binding agreement can be made today, for a deposit of £1,000,000 to be made at Time 2 periods into the future, which will return:

£1,000,000 x 1.03

= £1,030,000 at Time 3 (periods into the future).

Applications

A common application of forward yields is the pricing of forward rate agreements.

The forward yield is also known as the Forward rate or (sometimes) the Forward forward rate.

(The forward forward rate is technically slightly different, strictly referring to physical borrowings or deposits, rather than to derivative contracts.)

Quotation basis

Rates are generally quoted in wholesale markets as nominal annual rates.

Conversion

If we know the forward yield, we can calculate both the zero coupon yield and the par yield for the same maturities and risk class.

The conversion process and calculation stems from the 'no-arbitrage' relationship between the related yield curves.

This is illustrated on the page Converting from forward rates.

Notation

Notation varies between practitioners and contexts.

The yield conversion pages in this wiki use the following notation:

Periodic forward yields (f):

f_0-1: the rate per period for the maturity starting now and ending one period in the future.

f_1-2: the rate per period for the maturity starting one period in the future, and ending two periods in the future.

f_2-3: the rate per period for the maturity starting two periods in the future, and ending three periods in the future.

f_1-3: the rate per period for the maturity starting one period in the future, and ending three periods in the future.

And so on.

Periodic zero coupon yields (z):

z_0-1: the rate per period for the maturity starting now and ending one period in the future.

z_0-2: the rate per period for the maturity starting now, and ending two periods in the future, with all of the rolled up compounded interest paid at the end of period 2.

And so on.

It is best always to spell out expressly what cash flow pattern, maturity and quotation basis you intend, rather than assuming or hoping that others are familiar with your particular organisation's preferred notation.

@@ Line 1: / Line 1: @@
-The rate of return in the market today for a notional or actual deposit or borrowing:
+The fixed interest rate in the market today for an investment or borrowing commitment:
 #Starting at a fixed future date; and
 #Ending on a later fixed future date.
-'''Example 1'''
+The commitment can relate to a physical deposit or borrowing, or - more commonly - a derivative contract to be settled by reference to a notional deposit or borrowing.
-The forward yield for the maturity 2-3 periods is 3% per period.
+For example a forward rate might be quoted for a [[forward rate agreement]] for the maturity 2-5 months in the future.
-This means that a deposit of £1,000,000 at Time 2 periods would return:
-£1,000,000 x 1.03
-= £1,030,000 at Time 3 periods.
+===<span style="color:#4B0082">Example</span>===
+Taking a simpler example, say the forward rate (or forward yield) for the maturity 2-3 periods is 3% per period.
-A common application of forward yields is the pricing of forward rate agreements.
+is the time from today (into the future) when the investment or borrowing will start.
+is the time from today when the investment or borrowing will end.
-The forward yield is also known as the [[Forward rate]] or (sometimes) the Forward forward rate.
-(The [[forward forward rate]] is technically slightly different.)
+The difference between Time 3 periods and Time 2 periods is the length of the investment or borrowing.
+In this case the length of the investment or borrowing is 3 - 2 = 1 period.
-'''Conversion'''
-If we know the forward yield, we can calculate both the [[zero coupon yield]] and the [[par yield]] for the same maturities and risk class.
-The conversion process and calculation stems from the '[[no-arbitrage]]' relationship between the related yield curves.
+Assuming a deposit, 3% is the rate payable for period 3 only - a single period - which is pre-agreed today, 2 periods before the deposit is contracted to change hands.
+This means a mutually binding agreement can be made today, for a deposit of £1,000,000 to be made at Time 2 periods into the future, which will return:
-'''Example 2'''
+£1,000,000 x 1.03
-Periodic forward yields ('''f''') are:
+= £1,030,000 at Time 3 (periods into the future).
-f<sub>0-1</sub> = 0.02 per period (2%)
-f<sub>1-2</sub> = 0.04 per period (4%)
+===Applications===
+A common application of forward yields is the pricing of forward rate agreements.
-The total accumulated cash at Time 2 periods hence, from investing £1m at Time 0 is:
-£1m x 1.02 x 1.04
+The forward yield is also known as the [[Forward rate]] or (sometimes) the Forward forward rate.
-= £'''1.0608m'''
+(The [[forward forward rate]] is technically slightly different, strictly referring to physical borrowings or deposits, rather than to derivative contracts.)
-Under no-abitrage pricing conditions, the identical cash flows arise from investing in an outright zero coupon investment of two periods maturity, at the rate of '''z<sub>0-2</sub>''' per period, as follows:
+===Quotation basis===
-£1m x (1 + z<sub>0-2</sub>)<sup>2</sup> = £'''1.0608m'''
+Rates are generally quoted in wholesale markets as [[nominal annual rate]]s.
-Using this information, we can calculate the zero coupon rate for two periods' maturity.
+===Conversion===
+If we know the forward yield, we can calculate both the [[zero coupon yield]] and the [[par yield]] for the same maturities and risk class.
-(1 + z<sub>0-2</sub>)<sup>2</sup> = 1.0608
+The conversion process and calculation stems from the '[[no-arbitrage]]' relationship between the related yield curves.
-+ z<sub>0-2</sub> = 1.0608<sup>(1/2)</sup>
-z<sub>0-2</sub> = 1.0608<sup>(1/2)</sup> - 1
+This is illustrated on the page [[Converting from forward rates]].
-= 0.029951 per period (= 2.9951%)
+===Notation===
-Investing the same £1m in the two-periods maturity zero coupon instrument on these terms would return the same terminal cash flow of £1.0608m as the forward investments, as follows:
+Notation varies between practitioners and contexts.
-£1m x (1.029951)<sup>2</sup>
+The yield conversion pages in this wiki use the following notation:
-= £'''1.0608m'''
+''Periodic forward yields ('''f'''):''
-'''Example 3'''
+f<sub>0-1</sub>: the rate per period for the maturity starting now and ending one period in the future.
-Now using the zero coupon rates ('''z'''), the par rates ('''p''') can also be calculated in turn.
+f<sub>1-2</sub>: the rate per period for the maturity starting one period in the future, and ending two periods in the future.
+f<sub>2-3</sub>: the rate per period for the maturity starting two periods in the future, and ending three periods in the future.
-The periodic zero coupon yields ('''f''') are:
+f<sub>1-3</sub>: the rate per period for the maturity starting one period in the future, and ending three periods in the future.
-z<sub>0-1</sub> = 0.02 per period (2%)
+And so on.
-z<sub>0-2</sub> = 0.029951 per period (2.9951%)
+''Periodic zero coupon yields ('''z'''):''
-The no-arbitrage relationship between par rates and zero coupon rates is summarised in the formula:
+z<sub>0-1</sub>: the rate per period for the maturity starting now and ending one period in the future.
-p<sub>0-n</sub> = (1 - DF<sub>n</sub>) / CumDF<sub>n</sub>
+z<sub>0-2</sub>: the rate per period for the maturity starting now, and ending two periods in the future, with all of the rolled up compounded interest paid at the end of period 2.
+And so on.
-''Where:''
-p<sub>0-n</sub> = the par rate for maturity n periods, starting now
+It is best always to spell out expressly what cash flow pattern, maturity and quotation basis you intend, rather than assuming or hoping that others are familiar with your particular organisation's preferred notation.
-DF<sub>n</sub>) = the discount factor for 'n' periods maturity, calculated from the zero coupon rate (z<sub>n</sub>)
-CumDF<sub>n</sub>) = the total of the discount factors for maturities 1 to 'n' periods maturity, again calculated from the zero coupon rates (z<sub>1</sub> to z<sub>n</sub>)
-''Applying the formula:''
-p<sub>0-2</sub> = (1 - DF<sub>2</sub>) / CumDF<sub>2</sub>
-p<sub>0-2</sub> = (1 - 1.029951<sup>-2</sup>) / (1.02<sup>-1</sup> + 1.029951<sup>-2</sup>)
-= 0.029803 (= 2.9803% per period)
-This is the fair (no-arbitrage) market price for the par instrument, which will produce the identical terminal cash flow of £1.0608m as follows:
-Cash flows from the two period par instrument, paying periodic interest of 2.9803% per period, assuming an initial investment of £1m:
-Interest coupon at Time 1 period = £1m x 0.029803 = £0.029803m
-Principal + interest at Time 2 periods = £1m + 0.029803m = £1.029803m
-The coupon receivable at Time 1 period is reinvested at the pre-agreed forward rate of 4% (0.04) for the maturity 1-2 periods.
-So the Time 2 proceeds from the reinvested coupon received at Time 1 are:
-£0.029803 x 1.04
-= £0.030995 at Time 2
-The total terminal value at Time 2 periods is:
-.030995 + 1.029803
-= £'''1.0608m''' (as before)
 == See also ==
+* [[Converting from forward rates]]
+* [[Coupon]]
+* [[Discount factor]]
+* [[Falling yield curve]]
+* [[Flat yield curve]]
+* [[Forward rate agreement]]
+* [[Negative yield curve]]
+* [[No arbitrage conditions]]
+* [[Nominal annual rate]]
+* [[Par yield]]
+* [[Periodic yield]]
+* [[Positive yield curve]]
+* [[Rising yield curve]]
 * [[Yield curve]]
-* [[Par yield]]
 * [[Zero coupon yield]]
-* [[Forward rate agreement]]
-* [[Periodic yield]]
+[[Category:Corporate_financial_management]]
-* [[Discount factor]]
+[[Category:Manage_risks]]
-* [[Coupon]]

Forward yield: Difference between revisions

Latest revision as of 20:33, 1 July 2022

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