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)]Â¼ - 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

Â

 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

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

S3f5 = .027916

Doubling this rate gives you a rate of 5.58%

Â

Measuring Interest Rate Risk

Related Articles
1. Investing

### Explaining the Spot Rate

The spot rate is the immediate purchase price posted on exchanges for purchasing commodities, currency and securities.

### Using Interest Rate Parity To Trade Forex

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

### How To Lock In An Exchange Rate

Currency risk can be effectively hedged by locking in an exchange rate through the use of currency futures, forwards, options, or exchange-traded funds.

### Combining Forex Spot And Futures Transactions

The spot, futures and option currency markets can be traded together for maximum downside protection and profit.

### What Does Spot Price Mean?

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

### 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.

### Managing Interest Rate Risk

Interest rate risk stems from the possibility that an interest-bearing assetâ€™s value will change due to changing interest rates.
8. Insights

### 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. Personal Finance

### How Interest Rate Cuts Affect Consumers

Traders rejoice when the Fed drops the rate, but is it good news for all? Find out here.
1. ### Short Selling, or Selling Something You Don't Own

Money can be made without actually owning any shares, but short selling isn't for new investors.
2. ### Determining a Firm's Percentage of Credit Sales

Find out where to look for information about determining a company's percentage of credit sales.
3. ### How Did Kidder Peabody's Joseph Jett Lose \$350M?

The 1980s were a rough decade for Kidder, Peabody & Co. thanks to bond trader Joseph Jett.
4. ### What Is a Blank-Check Company?

A blank-check company has a business plan based on a merger or acquisition with another company.