Let's say an investor buys a two-year zero-coupon bond. The proceeds will equal:
X (1 + z4)4.

The investor could also buy a six-month Treasury bill and reinvest the proceeds every six months for two years. In this case, the value would be:

X (1 + z1)(1 + future rate at time 2)(1+ future rate at time 3) (1 + future rate at time 4)

Because these two investments must be equal this tells us that:

X (1 + z4)4 = X (1 + z1)(1 + future rate at time 2)(1+ future rate at time 3)(1 + future rate at time 4)

So Z4 = [(1 + z1)(1 + future rate at time 2)(1+ future rate at time 3)(1 + future rate at time 4)]¼ - 1

This equation states that the two-year spot rate depends on the current six-month rate and the following three six-month spot rates.

As we can see, short-term forward rates must equal spot rates or else an arbitrage opportunity can exist in the market place.

Compute Spot Rates if Given Forward Rates, and Forward Rates if Given Spot Rates
Computing a forward rate by using spot rates is covered above. Using spot rates, an investor can develop any forward rate.

There are two elements to the forward rate. The first is when the future rate begins. The second is the length of time for that rate. The notation is length of time of the forward rate f when the forward rate began. For example, a 2 f 8 would be the 1-year (two six-month periods) forward rate beginning four years (eight six-month periods) from now.

To solve for tFm use the following equation:

Formula 15.13

 

tFm =[ (1 + Zm+t)m+t / (1 + Zm)m] 1/t - 1

So for a 3f5 it would equal an equation of: [(1 + z8)8/ (1 + z5)5]1/3 -1

Example:
Z3(the 1.5 year spot rate) = 3.5%/2 = .0175
Z5 (the 2.5 year spot rate) = 4.25%/2 = .02125

Answer:
So 3f5 =[(1.02125)/ (1.0175)5]1/3 -1

S3f5 = .027916

Doubling this rate gives you a rate of 5.58%

 

Measuring Interest Rate Risk

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