8.3 Using It To Determine Forward Rates

Problem. Given the yield curve as published by the financial press, calculate the implied forward rates at

all maturities.

Solution Strategy. We will use the yield curve that you entered in a spreadsheet for The Yield Curve -

Obtaining It From Bond Listings. We will calculate the forward rates implied by the yield curve and

then graph our results.

FIGURE 8.3 Spreadsheet Model of The Yield Curve – Using It To Determine Forward Rates.

How To Build This Spreadsheet Model.

1. Start with the Bond Listings Spreadsheet. Open the spreadsheet that you created for The Yield

Curve - Obtaining It From Bond Listings and immediately save the spreadsheet under a new

name using the File | Save As command.

2. Insert a Column and Format It. Select the cell E1 and click on Insert | Columns. To get rid of

the yellow background, select the range E5:E17, click on Format | Cells, click on the Patterns

tab, click on the No Colors button, and click on OK.

3. Forward Rates. The forward rate from date T-1 to date T is given by



1, 1

1

1

1

T

T

T T T

T

k

FR

r −−

−







,

where T k is the date T yield and T 1 k −is the date T-1 yield. More generally, the forward rate from

any date t to date T is given by



,

1

1

1

T

T t T

t T t

t

k

FR

k

−





.

Solving for the forward rate, we obtain





1

( ) 1

1

1

T T t

T

T t t

t

k

FR

k

−

−



−



.

Enter =(((1+D5)^C5)/((1+D4)^C4))^(1/(C5-C4))-1 in cell E5 and copy it down.

4. Add The Forward Rates To The Graph. To add the forward rates, select the range E5:E17.,

click on Edit | Copy, then select the graph by clicking anywhere on the graph, and click on

Edit | Paste.

Using the forward rates as at least a rough forecast of future interest rates and taking the forward rates at

face value, they would suggest that interest rates are going to be in the 6% range in the short run, rising to

7% in five years, and declining below 5% in the long run. One difficulty with taking this interpretation

literally has to do with market segmentation in the demand for treasury securities. There is significantly

more demand for short-term bonds than bonds of other maturities, for their use in short-term cash

management. There is also extra demand by institutional bond funds for the newly-issued, longest

maturity treasury bond (the so-called, "on-the-run" bond). High demand means high prices, which means

low yields. Thus, the yield curve is typically has lower yields at the short end and the long end due to this

segmentation in market demand. It is not clear whether this yield curve would be nearly flat or not in the

absence of market segmentation. Ignoring the extreme forward rates generated by the short run and long

run segmentation, the forecast seems to be between 6.0% and 6.5%.

Problem. Given the yield curve as published by the financial press, calculate the implied forward rates at

all maturities.

Solution Strategy. We will use the yield curve that you entered in a spreadsheet for The Yield Curve -

Obtaining It From Bond Listings. We will calculate the forward rates implied by the yield curve and

then graph our results.

FIGURE 8.3 Spreadsheet Model of The Yield Curve – Using It To Determine Forward Rates.

How To Build This Spreadsheet Model.

1. Start with the Bond Listings Spreadsheet. Open the spreadsheet that you created for The Yield

Curve - Obtaining It From Bond Listings and immediately save the spreadsheet under a new

name using the File | Save As command.

2. Insert a Column and Format It. Select the cell E1 and click on Insert | Columns. To get rid of

the yellow background, select the range E5:E17, click on Format | Cells, click on the Patterns

tab, click on the No Colors button, and click on OK.

3. Forward Rates. The forward rate from date T-1 to date T is given by



1, 1

1

1

1

T

T

T T T

T

k

FR

r −−

−







,

where T k is the date T yield and T 1 k −is the date T-1 yield. More generally, the forward rate from

any date t to date T is given by



,

1

1

1

T

T t T

t T t

t

k

FR

k

−





.

Solving for the forward rate, we obtain





1

( ) 1

1

1

T T t

T

T t t

t

k

FR

k

−

−



−



.

Enter =(((1+D5)^C5)/((1+D4)^C4))^(1/(C5-C4))-1 in cell E5 and copy it down.

4. Add The Forward Rates To The Graph. To add the forward rates, select the range E5:E17.,

click on Edit | Copy, then select the graph by clicking anywhere on the graph, and click on

Edit | Paste.

Using the forward rates as at least a rough forecast of future interest rates and taking the forward rates at

face value, they would suggest that interest rates are going to be in the 6% range in the short run, rising to

7% in five years, and declining below 5% in the long run. One difficulty with taking this interpretation

literally has to do with market segmentation in the demand for treasury securities. There is significantly

more demand for short-term bonds than bonds of other maturities, for their use in short-term cash

management. There is also extra demand by institutional bond funds for the newly-issued, longest

maturity treasury bond (the so-called, "on-the-run" bond). High demand means high prices, which means

low yields. Thus, the yield curve is typically has lower yields at the short end and the long end due to this

segmentation in market demand. It is not clear whether this yield curve would be nearly flat or not in the

absence of market segmentation. Ignoring the extreme forward rates generated by the short run and long

run segmentation, the forecast seems to be between 6.0% and 6.5%.