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.
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.
Example 1: Forward to zero coupon rates
Periodic forward yields (f) are:
f0-1 = 0.02 per period (2%)
f1-2 = 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
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 z0-2 per period, as follows:
£1m x (1 + z0-2)2 = £1.0608m
Using this information, we can now calculate the zero coupon yield for two periods' maturity.
(1 + z0-2)2 = 1.0608
1 + z0-2 = 1.0608(1/2)
z0-2 = 1.0608(1/2) - 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)2
This is the same result as enjoyed from the forward investments, as expected.
Example 2: Forward to par rates
Now using the zero coupon rates (z), the par rates (p) can also be calculated in turn.
The periodic zero coupon yields (z) are:
z0-1 = 0.02 per period (2%)
z0-2 = 0.029951 per period (2.9951%)
The no-arbitrage relationship between par rates and zero coupon rates is summarised in the formula:
p0-n = (1 - DFn) / CumDFn
p0-n = the par rate for maturity n periods, starting now
DFn = the discount factor for 'n' periods maturity, calculated from the zero coupon rate (zn)
CumDFn = the total of the discount factors for maturities 1 to 'n' periods maturity, again calculated from the zero coupon rates (z1 to zn)
Applying the formula:
p0-2 = (1 - DF2) / CumDF2
p0-2 = (1 - 1.029951-2) / (1.02-1 + 1.029951-2)
= 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 = £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
= £0.030995m 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.
- Forward yield
- Yield curve
- Zero coupon yield
- Par yield
- Forward rate agreement
- Periodic yield
- Discount factor
- Flat yield curve
- Rising yield curve
- Falling yield curve
- Positive yield curve
- Negative yield curve
- Converting from zero coupon rates
- Converting from par rates