Suppose you are contemplating between purchasing one two$-$year bond today versus purchasing two consecutive one$-$year bonds. Based $on$ the video, which $of$ the following describes what would make you indifferent between the two options?

The average $of$ what a one$-$year bond pays today and what you expect a one$-$year bond will pay $in$ one year from today $is$ equal $to$ what a two$-$year bond pays today.

What a two$-$year bond pays today $is$ equal $to$ the sum $of$ what two one$-$year bonds pay one year from now.

The sum $of$ what a one$-$year bond pays today and what you expect a one$-$year bond will pay $in$ one year from today $is$ equal $to$ what a two$-$year bond pays today.

The sum $of$ what two one$-$year bonds pay today $is$ equal $to$ what a two$-$year bond pays today.

Suppose that $a1-$year Treasury bond currently yields $5\mathrm{.}00\%,$ and $a2-$year bond yields $5\mathrm{.}50\%.Asan$ investor, you have two options:

Option $1$: Buy $a2-$year security and hold $it$ for $2$ years.

Option $2$: Buy $a1-$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=$$$1(1+0\mathrm{.}055)2=$$$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=$$$1(1+0\mathrm{.}05)(1+x)=$$$1\mathrm{.}113,$

where $x$

stands for the expected interest rate $ona1-$year Treasury security $1$ year from now.

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

$=$

$$1\mathrm{.}113025$

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

$=$

$$1\mathrm{.}113025$

$(1+x)$

$=$

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

$x$

$=$

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

$=$

$0\mathrm{.}0600238,or6\mathrm{.}00238\%$

Suppose your friend $is$ deciding between investing $in$ two consecutive $1-$year Treasury bonds and $a2-$year Treasury bond. The yield $ona1-$year bond $is4\mathrm{.}70\%$ today and the yield $ona2-$year bond $is6\mathrm{.}00\%.$ You tell your friend that $if$ the expected interest rate $ona1-$year bond $1$ year from now $is7\mathrm{.}32\%,$ 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 $isno$ maturity risk premium.

Security

Yield

$(Percent)$

$1-$year $4\mathrm{.}70$

$2-$year $6\mathrm{.}00$

$3-$year $7\mathrm{.}10$

$4-$year $8\mathrm{.}30$

$If$ you want $to$ forecast the yield $ona1-$year security $in$ one year from now, you need $to$ build the following equation:

$=.$ And the yield $on1-$year security $in$ one year would $be.$

$If$ you want $to$ forecast the yield $ona1-$year security two years from now, you need $to$ build the following equation:

$=.$ And the yield $on1-$year security $in$ two years would $be.$

$If$ you want $to$ forecast the yield $ona2-$year security one year from now, you need $to$ build the following equation:

$=.$ And the yield $on2-$year security $in$ one year would $be.$

$If$ you want $to$ forecast the yield $ona3-$year security one year from now, you need $to$ build the following equation:

$=.$ And the yield $on3-$year security $in$ one year would $be.$

Step $3$: Practice: Pure Expectations Theory

Now $its$ time for you $to$ practice what you$ve$ learned.

Suppose the market offers the following Treasury $securities$:

Treasury security

Yield

$(Percent)$

$1-$year $4\mathrm{.}70$

$2-$year $6\mathrm{.}00$

$3-$year $7\mathrm{.}10$

$4-$year $8\mathrm{.}30$

$5-$year $9\mathrm{.}80$

$6-$year $11\mathrm{.}50$

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