imported>Doug Williamson |
imported>Doug Williamson |
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| 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:
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| #Starting at a fixed future date; and
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| #Ending on a later fixed future date.
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| | Basis points, 0.01% per point for interest rates. |
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| The forward rate is also known as the [[forward yield]].
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| '''Conversion'''
| | Forward foreign exchange rate points. |
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| 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.
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| The conversion process and calculation stems from the '[[no-arbitrage]]' relationship between the related yield curves. This means - for example - 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.
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| | Foreign exchange swap rate points. |
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| <span style="color:#4B0082">'''Example 1: Forward to zero coupon rates'''</span>
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| Periodic forward yields ('''f''') are:
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| f<sub>0-1</sub> = 0.02 per period (2%)
| | ''US.'' |
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| f<sub>1-2</sub> = 0.04 per period (4%)
| | 1% of a loan principal amount. |
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| The total accumulated cash at Time 2 periods hence, from investing £1m at Time 0 is:
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| £1m x 1.02 x 1.04
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| = £'''1.0608'''m
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| 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:
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| £1m x (1 + z<sub>0-2</sub>)<sup>2</sup> = £'''1.0608'''m
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| Using this information, we can now calculate the zero coupon yield for two periods' maturity.
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| (1 + z<sub>0-2</sub>)<sup>2</sup> = 1.0608
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| 1 + z<sub>0-2</sub> = 1.0608<sup>(1/2)</sup>
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| z<sub>0-2</sub> = 1.0608<sup>(1/2)</sup> - 1
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| = '''0.029951''' per period (= 2.9951%)
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| 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.
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| 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.
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| Investing the same £1m in a two-periods maturity zero coupon instrument at a rate of 2.9951% per period would return:
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| £1m x (1.029951)<sup>2</sup>
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| = £'''1.0608'''m
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| ''This is the same result as enjoyed from the forward investments, as expected.''
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| <span style="color:#4B0082">'''Example 2: Forward to par rates'''</span>
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| Now using the zero coupon rates ('''z'''), the par rates ('''p''') can also be calculated in turn.
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| The periodic zero coupon yields ('''z''') are:
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| z<sub>0-1</sub> = 0.02 per period (2%)
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| z<sub>0-2</sub> = 0.029951 per period (2.9951%)
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| The no-arbitrage relationship between par rates and zero coupon rates is summarised in the formula:
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| p<sub>0-n</sub> = (1 - DF<sub>n</sub>) / CumDF<sub>n</sub>
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| ''Where:''
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| p<sub>0-n</sub> = the par rate for maturity n periods, starting now
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| DF<sub>n</sub> = the discount factor for 'n' periods maturity, calculated from the zero coupon rate (z<sub>n</sub>)
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| 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>)
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| ''Applying the formula:''
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| p<sub>0-2</sub> = (1 - DF<sub>2</sub>) / CumDF<sub>2</sub>
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| p<sub>0-2</sub> = (1 - 1.029951<sup>-2</sup>) / (1.02<sup>-1</sup> + 1.029951<sup>-2</sup>)
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| = 0.029803 (= 2.9803% per period)
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| This is the theoretical fair (no-arbitrage) market price for the par instrument.
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| 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.
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| In theory, such an investment - again of £1m - should produce the same terminal cash flow. Let's see.
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| Cash flows from the two period par instrument, paying periodic interest of 2.9803% per period, assuming an initial investment of £1m:
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| Interest coupon at Time 1 period = £1m x 0.029803 = £<u>0.029803</u>m
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| Principal + interest at Time 2 periods = £1m + 0.029803m = £'''1.029803'''m
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| 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.
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| So the Time 2 proceeds from the reinvested coupon received at Time 1 are:
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| £0.029803 x 1.04
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| = £'''0.030995'''m at Time 2
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| The total terminal value at Time 2 periods is:
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| 0.030995 + 1.029803
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| = £'''1.0608'''m (as before)
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| The par rate we have calculated is indeed consistent with the no-arbitrage pricing relationship.
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| == See also == | | == See also == |
| * [[Forward yield]] | | * [[Basis point]] |
| * [[Yield curve]]
| | * [[Forward points]] |
| * [[Zero coupon yield]]
| | * [[Pip]] |
| * [[Par yield]]
| | * [[Point]] |
| * [[Forward rate agreement]] | | * [[Swap points]] |
| * [[Periodic yield]] | |
| * [[Discount factor]] | |
| * [[Coupon]] | |
| * [[Flat yield curve]]
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| * [[Rising yield curve]]
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| * [[Falling yield curve]]
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| * [[Positive yield curve]]
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| * [[Negative yield curve]]
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| * [[Converting from zero coupon rates]]
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| * [[Converting from par rates]]
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| ===Other resources===
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| [[Media:2013_09_Sept_-_Simple_solutions.pdf| The Treasurer students, Simple solutions]]
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