Forward yield

The rate of return 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.

Example 1

The forward yield for the maturity 2-3 periods is 3% per period.

This means that a deposit of £1,000,000 at Time 2 periods would return:

£1,000,000 x 1.03

= £1,030,000 at Time 3 periods.

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.)

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.

Example 2

Periodic forward yields (f) are:

f_0-1 = 0.02 per period (2%)

f_1-2 = 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-abitrage pricing conditions, the identical cash flows arise from investing in an outright zero coupon investment of two periods maturity, at the rate of z_0-2 per period, as follows:

£1m x (1 + z_0-2)² = £1.0608m

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

(1 + z_0-2)² = 1.0608

1 + z_0-2 = 1.0608^(1/2)

z_0-2 = 1.0608^(1/2) - 1

= 0.029951 per period (= 2.9951%)

Investing the same £1m in the two-periods maturity zero coupon instrument on these terms would return the same terminal cash flow of £1.0608m as the forward investments, as follows:

£1m x (1.029951)²

= £1.0608m

Example 3

Now using the zero coupon rates (z), the par rates (p) can also be calculated in turn.

The periodic zero coupon yields (f) are:

z_0-1 = 0.02 per period (2%)

z_0-2 = 0.029951 per period (2.9951%)

The no-arbitrage relationship between par rates and zero coupon rates is summarised in the formula:

p_0-n = (1 - DF_n) / CumDF_n

Where:

p_0-n = the par rate for maturity n periods, starting now

DF_n) = the discount factor for 'n' periods maturity, calculated from the zero coupon rate (z_n)

CumDF_n) = the total of the discount factors for maturities 1 to 'n' periods maturity, again calculated from the zero coupon rates (z₁ to z_n)

Applying the formula:

p_0-2 = (1 - DF₂) / CumDF₂

p_0-2 = (1 - 1.029951^-2) / (1.02^-1 + 1.029951^-2)

= 0.029803 (= 2.9803% per period)

This is the fair (no-arbitrage) market price for the par instrument, which will produce the identical terminal cash flow of £1.0608m as follows:

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

= £0.030995 at Time 2

The total terminal value at Time 2 periods is:

0.030995 + 1.029803

= £1.0608m (as before)

See also

• Yield curve
• Par yield
• Zero coupon yield
• Forward rate agreement
• Periodic yield
• Discount factor
• Coupon