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.
-<span style="color:#4B0082">'''Example 1'''</span>
+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 is the rate payable for period 3 only - a single period - which is pre-agreed today, 2 periods before the deposit or borrowing is contracted to change hands.
-This means that 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
+===<span style="color:#4B0082">Example</span>===
-= £1,030,000 at Time 3 periods.
+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 difference between Time 3 periods and Time 2 periods is the length of the investment or borrowing.
-(The [[forward forward rate]] is technically slightly different.)
+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.
+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.
-The conversion process and calculation stems from the '[[no-arbitrage]]' relationship between the related yield curves. This means that the cash flows from a two-year '[[outright]]' deposit must be identical to the cash flows from a '[[synthetic]]' two-year deposit, built from a combination of forward deals.
+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
-<span style="color:#4B0082">'''Example 2'''</span>
+= £1,030,000 at Time 3 (periods into the future).
-Periodic forward yields ('''f''') are:
-f<sub>0-1</sub> = 0.02 per period (2%)
+===Applications===
-f<sub>1-2</sub> = 0.04 per period (4%)
+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:
+The forward yield is also known as the [[Forward rate]] or (sometimes) the Forward forward rate.
-£1m x 1.02 x 1.04
+(The [[forward forward rate]] is technically slightly different, strictly referring to physical borrowings or deposits, rather than to derivative contracts.)
-= £'''1.0608m'''
+===Quotation basis===
-Under no-arbitrage pricing conditions, the identical terminal cash flow of £1.0608m results from investing in an outright zero coupon investment of two periods maturity, at the market yield of '''z<sub>0-2</sub>''' per period, as follows:
+Rates are generally quoted in wholesale markets as [[nominal annual rate]]s.
-£1m x (1 + z<sub>0-2</sub>)<sup>2</sup> = £'''1.0608m'''
+===Conversion===
-Using this information, we can now calculate the zero coupon yield for two periods' maturity.
+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
-+ z<sub>0-2</sub> = 1.0608<sup>(1/2)</sup>
-z<sub>0-2</sub> = 1.0608<sup>(1/2)</sup> - 1
-= '''0.029951''' per period (= 2.9951%)
-This is the market zero coupon rate which we would enjoy if we were to make a two-year deposit, putting our money into the deposit today.
-The no-arbitrage relationship says that making such a deposit should produce the identical terminal cash flow of £1.0608m. Let's see if that's borne out by our calculations.
-Investing the same £1m in a two-periods maturity zero coupon instrument at a rate of 2.9951% per period would return:
-£1m x (1.029951)<sup>2</sup>
-= £'''1.0608m'''
-''This is the same result as enjoyed from the forward investments, as expected.''
-<span style="color:#4B0082">'''Example 3'''</span>
-Now using the zero coupon rates ('''z'''), the par rates ('''p''') can also be calculated in turn.
-The periodic zero coupon yields ('''z''') are:
-z<sub>0-1</sub> = 0.02 per period (2%)
-z<sub>0-2</sub> = 0.029951 per period (2.9951%)
-The no-arbitrage relationship between par rates and zero coupon rates is summarised in the formula:
-p<sub>0-n</sub> = (1 - DF<sub>n</sub>) / CumDF<sub>n</sub>
-''Where:''
-p<sub>0-n</sub> = the par rate for maturity n periods, starting now
-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)
+The conversion process and calculation stems from the '[[no-arbitrage]]' relationship between the related yield curves.
-This is the theoretical fair (no-arbitrage) market price for the par instrument.
-It is the calculated rate of interest payable on a two-year investment on par rate terms. This means that 2.9803% interest will be paid on the amount of the original investment, at Times 1 and 2 periods. In addition, the original investment will be repaid at Time 2.
+This is illustrated on the page [[Converting from forward rates]].
-In theory, such an investment - again of £1m - should produce the same terminal cash flow. Let's see.
+===Notation===
+Notation varies between practitioners and contexts.
-Cash flows from the two period par instrument, paying periodic interest of 2.9803% per period, assuming an initial investment of £1m:
+The yield conversion pages in this wiki use the following notation:
-Interest coupon at Time 1 period = £1m x 0.029803 = £<u>0.029803</u>m
+''Periodic forward yields ('''f'''):''
-Principal + interest at Time 2 periods = £1m + 0.029803m = £'''1.029803'''m
+f<sub>0-1</sub>: the rate per period for the maturity starting now and ending one period in the future.
+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.
-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.
+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.
-So the Time 2 proceeds from the reinvested coupon received at Time 1 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.
-£0.029803 x 1.04
+And so on.
-= £'''0.030995'''m at Time 2
+''Periodic zero coupon yields ('''z'''):''
-The total terminal value at Time 2 periods is:
+z<sub>0-1</sub>: the rate per period for the maturity starting now and ending one period in the future.
-.030995 + 1.029803
+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.
-= £'''1.0608m''' (as before)
+And so on.
-The par rate we have calculated is indeed consistent with the no-arbitrage pricing relationship.
+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.
 == See also ==
-* [[Yield curve]]
+* [[Converting from forward rates]]
-* [[Zero coupon yield]]
+* [[Coupon]]
+* [[Discount factor]]
+* [[Falling yield curve]]
+* [[Flat yield curve]]
+* [[Forward rate agreement]]
+* [[Negative yield curve]]
+* [[No arbitrage conditions]]
+* [[Nominal annual rate]]
 * [[Par yield]]
-* [[Forward rate agreement]]
 * [[Periodic yield]]
-* [[Discount factor]]
+* [[Positive yield curve]]
-* [[Coupon]]
-* [[Flat yield curve]]
 * [[Rising yield curve]]
-* [[Falling yield curve]]
+* [[Yield curve]]
-* [[Positive yield curve]]
+* [[Zero coupon yield]]
-* [[Negative yield curve]]
-* [[Converting from forward rates]]
+[[Category:Corporate_financial_management]]
+[[Category:Manage_risks]]

Forward yield: Difference between revisions

Latest revision as of 20:33, 1 July 2022

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