**CFA Level 1 - Fixed Income Investments**

Forward rates can be defined as the way the market is feeling about the future movements of interest rates. They do this by extrapolating from the risk-free theoretical spot rate. For example, it is possible to calculate the one-year forward rate one year from now. Forward rates are also known as implied forward rates.

To compute a bond's value using forward rates, you must first calculate this rate. After you have calculated this value, you just plug it into the formula for the prices of a bond where the interest rate or yield would be inserted.

An investor can purchase a one-year Treasury bill or buy a six-month bill and roll it into another six-month bill once it matures. The investor will be indifferent if they both produce the same result. An investor will know the spot rate for the six-month bill and the one-year bond, but he or she will not know the value of a six-month bill that is purchased six months from now. Given these two rates though, the forward rate on a six-month bill will be the rate that equalizes the dollar return between the two types of investments mentioned earlier.

An investor buys a six-month bill for $x. At the end of six months, the value would equal:

where z

F= one half the forward rate (expressed as a BEY) of a six-month rate six months from now. If he bought the six-month bill and reinvested the proceeds for another six months the dollar return would be calculated like this:

X(1 +z

For the one year investment the future dollars would be x(1 +z)

So F = (1 + z

Then double F to get the BEY.

Here are some numbers to try in this formula:

Six-month spot rate is 0.05 = 0.025 = z

1-year spot rate is 0.055 = 0.0275= z

F = ( 1.0275)

F = .030 or .06 or 6% BEY

To confirm this:

X(1.025)(1.03) = 1.05575

X(1.02575)

Once you have developed the future rate curve, you can continue to run and gun in the basic bond equation using the forward rates instead of the discount rate to value the bond.x(1 + z

**Example:**An investor can purchase a one-year Treasury bill or buy a six-month bill and roll it into another six-month bill once it matures. The investor will be indifferent if they both produce the same result. An investor will know the spot rate for the six-month bill and the one-year bond, but he or she will not know the value of a six-month bill that is purchased six months from now. Given these two rates though, the forward rate on a six-month bill will be the rate that equalizes the dollar return between the two types of investments mentioned earlier.

**Answer:**An investor buys a six-month bill for $x. At the end of six months, the value would equal:

where z

_{1}= one half of the bond equivalent yield on the six month spot rate.F= one half the forward rate (expressed as a BEY) of a six-month rate six months from now. If he bought the six-month bill and reinvested the proceeds for another six months the dollar return would be calculated like this:

X(1 +z

_{1}) (1 + F)For the one year investment the future dollars would be x(1 +z)

^{2}So F = (1 + z

_{2})^{2}/ (1 + z_{1}) - 1Then double F to get the BEY.

Here are some numbers to try in this formula:

Six-month spot rate is 0.05 = 0.025 = z

_{1}1-year spot rate is 0.055 = 0.0275= z

_{2}_{}F = ( 1.0275)

^{2}/ (1.025) -1F = .030 or .06 or 6% BEY

To confirm this:

X(1.025)(1.03) = 1.05575

X(1.02575)

^{2 }= 1.05575Once you have developed the future rate curve, you can continue to run and gun in the basic bond equation using the forward rates instead of the discount rate to value the bond.x(1 + z

_{1})
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