View source for Converting from forward rates

The forward rate is the rate of return - or cost of borrowing - contracted in the market today for a notional or actual deposit or borrowing: 
#Starting at a fixed future date; and 
#Ending on a later fixed future date.


The forward rate is also known as the [[forward yield]].


'''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 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.


<span style="color:#4B0082">'''Example 1'''</span>

Periodic forward yields ('''f''') are:

f<sub>0-1</sub> = 0.02 per period (2%)

f<sub>1-2</sub> = 0.04 per period (4%)


The total accumulated cash at Time 2 periods hence, from investing £1m at Time 0 is:

£1m x 1.02 x 1.04

= £'''1.0608m'''


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:

£1m x (1 + z<sub>0-2</sub>)<sup>2</sup> = £'''1.0608m'''


Using this information, we can now calculate the zero coupon yield for two periods' maturity.


(1 + z<sub>0-2</sub>)<sup>2</sup> = 1.0608

1 + 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 2'''</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)


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.


In theory, such an investment - again of £1m - should produce the same terminal cash flow. Let's see.


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 = £<u>0.029803</u>m

Principal + interest at Time 2 periods = £1m + 0.029803m = £'''1.029803'''m


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'''m at Time 2


The total terminal value at Time 2 periods is:

0.030995 + 1.029803

= £'''1.0608m''' (as before)


The par rate we have calculated is indeed consistent with the no-arbitrage pricing relationship.


== See also ==
* [[Forward yield]]
* [[Yield curve]]
* [[Zero coupon yield]]
* [[Par yield]]
* [[Forward rate agreement]]
* [[Periodic yield]]
* [[Discount factor]]
* [[Coupon]]
* [[Flat yield curve]]
* [[Rising yield curve]]
* [[Falling yield curve]]
* [[Positive yield curve]]
* [[Negative yield curve]]
* [[Converting from zero coupon rates]]
* [[Converting from par rates]]