# Converting from forward rates

The forward rate is the rate of return - or cost of borrowing - contracted in the market today for a notional or actual deposit or borrowing:

- Starting at a fixed future date; and
- Ending on a later fixed future date.

The forward rate is also known as the forward yield.

**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. This means - for example - that the cash flows from a two-year 'outright' deposit must be identical to the cash flows from a 'synthetic' two-year deposit, built from a combination of forward deals.

**Example 1: Forward to zero coupon rates**

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

= £**1.0608**m

Under no-arbitrage pricing conditions, the identical terminal cash flow of £1.0608m results from investing in an outright zero coupon investment of two periods maturity, at the market yield of **z _{0-2}** per period, as follows:

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

Using this information, we can now calculate the zero coupon yield 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%)

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}

= £**1.0608**m

*This is the same result as enjoyed from the forward investments, as expected.*

**Example 2: Forward to par rates**

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

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-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 = £__0.029803__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.

## See also

- Forward yield
- Yield curve
- Zero coupon yield
- Par yield
- Forward rate agreement
- Periodic yield
- Discount factor
- Coupon
- Flat yield curve
- Rising yield curve
- Falling yield curve
- Positive yield curve
- Negative yield curve
- Converting from zero coupon rates
- Converting from par rates