Suppose you are contemplating between purchasing one two$-$year bond today versus purchasing two consecutive one$-$year bonds. Based on the video, what determines the yield on a two$-$year bond?

The yield on the one$-$year bond plus what you expect the one$-$year bond to pay in one year.

What you expect two one$-$year bonds to pay in one year.

What you expect two one$-$year bonds to pay in two years.

The sum of yields on two one$-$year bond today.

Suppose that a $1-$year Treasury bond currently yields $5\mathrm{.}00\%,$ and a $2-$year bond yields $5\mathrm{.}50\%.$ As an investor, you have two options:

Option $1$: Buy a $2-$year security and hold it for $2$ years.

Option $2$: Buy a $1-$year security, hold it for $1$ year, and then at the end of the year reinvest the proceeds in another $1-$year security.

In two years, Option $1$ will yield the following amount per $$1$ you invested:

Yield at the end of year $2=$$$1\backslash$times $(1+0\mathrm{.}055)2=$$$1\mathrm{.}113$

The pure expectation theory implies that Option $2$ should yield the same amount, which can be expressed as follows:

Yield at the end of year $2=$$$1\backslash$times $(1+0\mathrm{.}05)\backslash$times $(1+$x$)=$$$1\mathrm{.}113,$

where x

stands for the expected interest rate on a $1-$year Treasury security $1$ year from now.

$$1\backslash$times $(1+0\mathrm{.}05)\backslash$times $(1+$x$)$

$=$

$$1\mathrm{.}113025$

$$1\mathrm{.}05\backslash$times $(1+$x$)$

$=$

$$1\mathrm{.}113025$

$(1+$x$)$

$=$

$$1\mathrm{.}113025$$$1\mathrm{.}05$

x

$=$

$$1\mathrm{.}113025$$$1\mathrm{.}051$

$=$

$0\mathrm{.}0600238,$ or $6\mathrm{.}00238\%$

Suppose your friend is deciding between investing in two consecutive $1-$year Treasury bonds and a $2-$year Treasury bond. The yield on a $1$year bond is $5\mathrm{.}50\%$ today and the yield on a $2-$year bond is $6\mathrm{.}60\%.$ You tell your friend that if the expected interest rate on a $1-$year bond $1$ year from now is$7\mathrm{.}71\%,$ then he should be indifferent between the two options.

Step $2$: Learn: Pure Expectations Theory

Watch the following video for an example, then answer the questions that follow.

Suppose you are given the yields on the following Treasury securities$.$ For simplicity, assume that there is no maturity risk premium.

Security

Yield

$($Percent$)$

$1$year $5\mathrm{.}50$

$2$year $6\mathrm{.}60$

$3$year $8\mathrm{.}10$

$4$year $9\mathrm{.}30$

If you want to forecast the yield on a $1$year security in one year from now, you need to build the following equation:

$=.$ And the yield on $1$year security in one year would be $.$

If you want to forecast the yield on a $1$year security two years from now, you need to build the following equation:

$=.$ And the yield on $1$year security in two years would be $.$

If you want to forecast the yield on a $2$year security one year from now, you need to build the following equation:

$=.$ And the yield on $2$year security in one year would be $.$

If you want to forecast the yield on a $3$year security one year from now, you need to build the following equation:

$=.$ And the yield on $3$year security in one year would be $.$

Step $3$: Practice: Pure Expectations Theory

Now it$$s time for you to practice what you$$ve learned.

Suppose the market offers the following Treasury securities:

Treasury security

Yield

$($Percent$)$

$1$year $5\mathrm{.}50$

$2$year $6\mathrm{.}60$

$3$year $8\mathrm{.}10$

$4$year $9\mathrm{.}30$

$5$year $10\mathrm{.}80$

$6$year $12\mathrm{.}70$

Make the necessary calculations and complete the following table using the data on the securities yields and the pure expectation theory.

Investment Yield

$1$year Treasury security, $1$ year from now

$2$year Treasury security, $2$ years from now

$3$year Treasury security, $1$ year from now

$4$year Treasury security, $2$ years from now