Let's say an investor buys a two-year zero-coupon bond. The proceeds will equal:
X (1 + z6)6.

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 1)(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 + z6)6 = X (1 + z1)(1+ future rate at time 1)(1 + future rate at time 2)(1+ future rate at time 3)

So Z6 = [(1 + z1)(1+ future rate at time 1)(1 + future rate at time 2)(1+ future rate at time 3)]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

Related Articles
  1. Trading

    Using Interest Rate Parity To Trade Forex

    Learn the basics of forward exchange rates and hedging strategies to understand interest rate parity.
  2. Investing

    The Money Market Hedge: How It Works

    Investopedia explains how to hedge foreign exchange risk using the money market, the financial market in which highly liquid and short-term instruments like Treasury bills, bankers’ acceptances ...
  3. Trading

    Why Forward Contracts Are The Foundation Of All Derivatives

    This article expands on the complex structure of derivatives by explaining how an investor can assess interest rate parity and implement covered interest arbitrage by using a currency forward ...
  4. Markets

    Calculating Future Value

    Future value is the value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today.
  5. Trading

    Managing Interest Rate Risk

    Interest rate risk stems from the possibility that an interest-bearing asset’s value will change due to changing interest rates.
  6. Investing

    Understanding the Spot Market

    A spot market is a market where a commodity or security is bought or sold and then delivered immediately.
  7. Investing

    What Does Spot Price Mean?

    Spot price is the current price at which a security may be bought or sold.
  8. Markets

    The International Fisher Effect: An Introduction

    The Fisher models have the ability to illustrate the expected relationship between interest rates, inflation and exchange rates.
  9. Investing

    Understanding The Time Value Of Money

    Find out why time really is money by learning to calculate present and future value.
  10. Trading

    Forex: Gauging Forex Market Sentiment With Open Interest

    Examining open interest on currency futures can help you confirm the strength of a trend in forex market sentiment.
Trading Center