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 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 3
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)
