Zero coupon yield
The rate of return on an investment today, for a single cashflow at the final maturity of the instrument. No intermediate interest is payable or receivable. (There are no interest coupons, hence the name 'zero coupon'.)
The zero coupon yield is equal to the current market rate of return on investments in zero coupon bonds of the same maturity.
Example 1: Cash flows from 3-period zero coupon instrument
The zero coupon yield for the maturity 0-3 periods is 2% per period.
This means that a deposit of £1,000,000 at Time 0 periods on these terms would return:
£1,000,000 x 1.023
= £1,061,208 at Time 3 periods.
(No intermediate interest is payable.)
An application of zero coupon yields is the pricing of zero coupon bonds.
The zero coupon yield is also known as the Zero coupon rate, spot rate, or spot yield.
Conversion
If we know the zero coupon yield, we can calculate both the forward 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 total cumulative cash flows from a two-year investment must be identical, whether the investment is built:
- 'Outright' from a two-year zero coupon investment
- Or as a synthetic deposit built using a forward contract, reinvesting intermediate principal and interest proceeds at a pre-agreed rate
- Or using a par investment, reinvesting intermediate interest to generate a total terminal cash flow
Example 2: Converting two-period zero coupon yields to forward yields
Periodic zero coupon yields (z) are:
z0-1 = 0.02 per period (2%)
z0-2 = 0.029951 per period (2.9951%)
The cash returned at Time 2 periods in the future, from investing £1m at Time 0 in a zero coupon instrument at a rate of 2.9951% per period, is:
£1m x 1.0299512
= £1.0608m
Under no-abitrage pricing conditions, the identical terminal cash flow of £1.0608m results from investing in an outright zero coupon investment of one periods maturity, together with a forward contract for the second period - for reinvestment at the forward market yield of f1-2 per period, as follows:
£1m x (1 + z0-1) x (1 + f1-2) = £1.0608m
Using this information, we can now calculate the forward yield for 1-2 periods' maturity.
1.02 x (1 + f1-2) = 1.0608
1 + f1-2 = 1.0608 / 1.02
f1-2 = (1.0608 / 1.02) - 1
= 1.04 - 1
= 0.04 per period (= 4%)
This is the market forward rate which we would enjoy if we were to pre-agree today, to make a one-period deposit, committing ourselves to put our money into the deposit one period in the future.
The no-arbitrage relationship says that making such a synthetic 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 this synthetic two-periods maturity zero coupon instrument would return:
After one period: £1m x 1.02 = £1.02m
Reinvested for the second period at the pre-agreed rate of 0.04 per period for one more period:
= £1.02m x 1.04
= £1.0608m
This is the same result as enjoyed from the outright zero coupon investment, as expected.
