# Converting from zero coupon rates

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

## No arbitrage conversion principles

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

## Zero coupon rate to forward rate conversion

**Example 1: Converting from one and 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}

= £**1.0608**m

Under no-arbitrage pricing conditions, the identical terminal cash flow of £1.0608m results from investing in an outright zero coupon investment of one period's 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}) = £**1.0608**m

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.

## Is terminal cash the same in each case?

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

= £**1.0608**m

*This is indeed the same result as enjoyed from the outright zero coupon investment, as expected.*

(The one period forward rate f_{0-1} represents the identical deal to the one period zero coupon rate z_{0-1}. For this reason the rate is also identical = 2% per period.)

## Zero coupon rates to par rates conversion

**Example 2: Converting from zero coupon rates to par rates**

Again using the given zero coupon rates (**z**), the par rates (**p**) can also be calculated.

The periodic zero coupon yields (**z**) 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 (z_{1} to z_{n})

*Applying the formula:*

p_{0-2} = (1 - DF_{2}) / CumDF_{2}

p_{0-2} = (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-period 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.

(The one period par rate p_{0-1} represents the identical deal to the one period zero coupon rate z_{0-1}. For this reason the rate is also identical = 2% per period.)

## Terminal cash is still the same

In theory, an investment of £1m in a par instrument should produce the same terminal cash flow as a zero coupon instrument or forward instruments. On these figures (forward rates of 2% and 4% for 0-1 and 1-2 periods maturity, respectively) we saw that was £1.0608m for an original investment of £1m.

Let's see if it's the same for the par instrument, assuming we arrange for the re-investment of any intermediate cash flows at today's forward rates.

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 = £__0.029803__m

Principal + interest at Time 2 periods = £1m + 0.029803m = £**1.029803**m

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**m at Time 2

The total terminal value at Time 2 periods is:

0.030995 + 1.029803

= £**1.0608**m (as before)

The par rate we have calculated is indeed consistent with the no-arbitrage pricing relationship.

## Present value is the same too

No-arbitrage pricing also says that the present value of the par instrument should be the same as the present value of the zero coupon and forward instruments.

Present values are calculated from discount factors based on the periodic zero coupon rates.

Continuing with our examples above, the zero coupon rates are:

0-1 period: 0.02 (2%) per period

0-2 periods: 0.029951 (2.9951%) per period

The related discount factors are:

1 period: 1.02^{-1}

2 periods: 1.029951^{-2}

The cash flows from the zero coupon instrument and the forward instrument are a cash inflow of £1.0608m at Time 2 periods.

The present value is:

£1.0608m x 1.029951^{-2}

= £**1.0000**m

**Example 3: A par bond trades at par**

A two-period par instrument pays periodic coupons of 2.9803%.

Prove that a £1m face value bond has a total present value of par (£1m), using the figures above.

*Solution*

Cash flows from the two period par instrument, paying periodic interest of 2.9803% per period, with a face value of £1m:

Coupon at Time 1 period = £1m x 0.029803 = £__0.029803__m

Principal + coupon at Time 2 periods = £1m + 0.029803m = £**1.029803**m

The present values of these cash flows are:

Time 1: £0.029803 x 1.02^{-1} = 0.02922

Time 2: £1.029803 x 1.029951^{-2} = 0.97078

Total = £**1.0000**m

The 'par bond' trades at par, as expected.

The pricing is consistent with no aribtrage pricing conditions.

## See also

- Bootstrap
- Converting from forward rates
- Converting from par rates
- Coupon
- Falling yield curve
- Flat yield curve
- Forward yield
- Negative yield curve
- No arbitrage conditions
- Par bond
- Par yield
- Positive yield curve
- Rising yield curve
- Spot rate
- Yield curve
- Zero
- Zero coupon bond
- Zero coupon yield