# Converting from zero coupon rates

The zero coupon rate is also known as the zero coupon yield, spot rate, or spot yield.

**Conversion**

If we know the zero coupon rates (yield curve) for a given risk class and set of maturities, we can calculate both the forward yields and the par yields 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 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 1: Converting two-period zero coupon yields to forward yields**

Periodic zero coupon yields (**z**) are:

z_{0-1} = 0.02 per period (2%)

z_{0-2} = 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 1.029951^{2}

= £**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 one periods maturity, together with a forward contract for the second period - for reinvestment at the forward market yield of **f _{1-2}** per period, as follows:

£1m x (1 + z_{0-1}) x (1 + f_{1-2}) = £**1.0608m**

Using this information, we can now calculate the forward yield for 1-2 periods' maturity.

1.02 x (1 + f_{1-2}) = 1.0608

1 + f_{1-2} = 1.0608 / 1.02

f_{1-2} = (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.*