Constructing Forward-Forward Yields from Zero-Coupon Yields in Excel

A zero-coupon yield is the annualized return of a bond that pays no interest and only returns the principal at maturity. A zero-coupon bond has a single cash flow at the end of its term. A forward-forward yield is the annualized return of a bond that starts at some future date and matures at another future date. A forward-forward bond has two cash flows: one at the start date and one at the end date.

To construct a forward-forward yield from zero-coupon yields, we need to use the concept of present value. Present value is the current worth of a future cash flow, discounted by an interest rate. The interest rate used to discount the cash flow is called the discount factor. The discount factor depends on the time period and the zero-coupon yield for that period.

For example, suppose we have the following zero-coupon yields for different time periods:

Table

Time period	Zero-coupon yield
1 year	3%
2 years	3.25%
3 years	3.35%

The discount factor for each time period is calculated by dividing 1 by 1 plus the zero-coupon yield. For example, the discount factor for 1 year is 1 / (1 + 0.03) = 0.9709. The discount factor for 2 years is 1 / (1 + 0.0325) = 0.9380. The discount factor for 3 years is 1 / (1 + 0.0335) = 0.9057.

Now, suppose we want to construct a forward-forward yield for a bond that starts in 2 years and matures in 3 years. This bond has two cash flows: one at the start date (2 years from now) and one at the end date (3 years from now). To find the forward-forward yield, we need to find the discount factor that equates the present value of these two cash flows. In other words, we need to solve for the discount factor x such that:

x * (cash flow at start date) = (discount factor for 3 years) * (cash flow at end date)

If we assume that the bond has a face value of 100 and pays no interest, then the cash flow at the start date is -100 and the cash flow at the end date is 100. Plugging these values into the equation, we get:

x * (-100) = (0.9057) * (100)

Solving for x, we get:

x = -0.9057

The forward-forward yield is the annualized rate that corresponds to this discount factor. To find it, we need to raise the discount factor to the power of 1 divided by the time period, and subtract 1. The time period is the difference between the end date and the start date, which is 1 year in this case. Therefore, the forward-forward yield is:

(-0.9057)^(1 / 1) – 1 = -0.1043

This means that the annualized return of the bond that starts in 2 years and matures in 3 years is -10.43%. This is a negative yield, which means that the bond is trading at a premium. The bond is more expensive than the zero-coupon bonds for the same time periods, because it locks in a lower interest rate for the future.

Basic Theory:

The zero-coupon yield curve provides yields for different maturities. Forward-forward yields, on the other hand, represent yields for future periods within the total maturity. The formula for calculating forward-forward yields from zero-coupon yields is:

$F_{t_1, t_2} = \left(\frac{1 + Y_{t_2}}{1 + Y_{t_1}}\right)^{\frac{t_2 - t_1}{t_2 - t_0}} - 1$

Where:

$F_{t_1, t_2}$ is the forward-forward yield between periods $t_1$ and $t_2$
$Y_{t_1}$ and $Y_{t_2}$ are the zero-coupon yields for periods $t_1$ and $t_2$
$t_0$ is the current time

Procedures:

Gather zero-coupon yields for different maturities.
Use the formula to calculate forward-forward yields between desired time periods.

Example Scenario:

Let’s consider the following zero-coupon yields for three different time periods: 1 year ( $Y_1 = 3\%$ ), 2 years ( $Y_2 = 4\%$ ), and 3 years ( $Y_3 = 5\%$ ). We want to find the forward-forward yield between year 1 and year 3.

Excel Calculation:

Create an Excel table with columns for time periods and zero-coupon yields.

Time Period	Zero-Coupon Yield
1	3%
2	4%
3	5%

Use the following Excel formula to calculate the forward-forward yield between year 1 and year 3:

$((1 + C3) / (1 + B3))^((A3 - A2) / (A3 - A1)) - 1$

Where:

A3 is the cell containing the maturity of year 3,
B3 is the cell containing the zero-coupon yield for year 1,
C3 is the cell containing the zero-coupon yield for year 3.

The result will be the forward-forward yield between year 1 and year 3.

Result:

The forward-forward yield between year 1 and year 3, based on the provided zero-coupon yields, is approximately $6.02\%$ .

Alternative Approaches:

Interpolation: If zero-coupon yields are not available for the exact time periods, linear interpolation can be used to estimate the missing values.
Bootstrapping: This involves constructing the zero-coupon yield curve by iteratively solving for unknown yields using bond prices for different maturities.