Arbitrage with Different Assets

The Term Structure of Interest Rates We just observed that there are many different assets and thus many interest rates in the economy. But these interest rates are all linked to each other because the same people (particularly banks and other financial institutions) trade in many different markets. One way in which assets differ is in terms of their maturity. To see how the returns on assets of different maturity are linked, consider two government bonds of different maturities: one- year bonds and two-year bonds. Here are two different ways you could save for two years.

1. Buy a one-year government bond. Collect the payment at the end of the year and then

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reinvest that money in another one-year bond. 2. Buy a two-year government bond.

There are three interest rates relevant to your choice. The first one is the current interest rate on a one-year bond. The second is the interest rate on a one-year bond next year. The third interest rate is the annualized nominal interest rate on a two-year government bond. An annualized interest rate is the interest rate earned each year on a loan that lasts many years, and the annualized interest factor is (1 + the annualized interest rate). For example, suppose that the annualized rate on a two-year loan is 6 percent. Then you would earn 6 percent per year for two years, and repayment after two years = $100 × 1.06 × 1.06 = $112.36. As you might expect, these three interest rates are connected, and we can understand how by again thinking about arbitrage. If you purchase the two-year government bond return, you get 100 × (annualized nominal interest factor on two-year bond)2. Conversely, if you purchase the two one-year bonds, you get 100 × (nominal interest factor this year) × (expected nominal interest factor next year). Notice that we have referred to next year’s interest factor as “expected.” This is because when you make your decision, you do not know what the interest rate will be. When (annualized nominal interest factor on two-year bond)2 = nominal interest factor this year× expected nominal interest factor next year, the two transactions have the same return. Once again, we can appeal to an arbitrage argument to say that we expect this equation to hold. There is one twist, however. When you make this decision, you do not know for sure what the interest rate will be on one-year bonds next year. You have to make a guess. Thus this arbitrage involves some risk.