# 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:

1. Starting at a fixed future date; and
2. 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 - 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

= £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 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

= £1.0608m

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

Where:

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.