$A$

$n$ investor $is$ considering the purchase $of$ zero$-$

coupon $U.S.$ Treasury bonds. $A30-$year zero$-$

coupon bond yielding $8\%$ can $be$ purchased today for

$$9\mathrm{.}94.At$ the end $of30$ years, the owner $of$ the bond

will receive $$100.$ The yield $of$ the bond $is$ related $to$

its price $by$ the following equation:

$P=\frac{100}{(1+y{)}^t}$

Here, $Pis$ the price $of$ the bond, $yis$ the yield $of$ the

bond, and $tis$ the maturity $of$ the bond measured $in$

years. Evaluating this equation for $t=30$ and $y=$

$0\mathrm{.}08$ gives $P=9\mathrm{.}94.$

The investor $is$ planning $to$ purchase a bond

today and sell $it$ one year from now. The investor $is$

interested $in$ evaluating the return $on$ the investment

$in$ the bond. Suppose, for example, that the yield $of$

the bond one year from now $is8\mathrm{.}5\%.$ Then the price

$of$ the bond one year later will $be$

$\{$:$0\mathrm{.}085{)}^{29}.$ The time remaining $to$ maturity $ist=29$

because one year has passed. The return for the

year $is-5\mathrm{.}54\%[=\frac{9\mathrm{.}39-9\mathrm{.}94}{9\mathrm{.}94}].$

$In$ addition $to$ the $30-$year$-$maturity zero$-$coupon

bond, the investor $is$ considering the purchase $of$

zero$-$coupon bonds with maturities $of2,5,10,or$

$20$ years. All $of$ the bonds are currently yielding

$8\mathrm{.}0\%.(Bond$ investors describe this $as$ a flat yield

curve.$)$ The investor cannot predict the future yields

$of$ the bonds with certainty. However, the investor

believes that the yield $of$ each bond one year from

now can $be$ modeled $by$ a normal distribution with

mean $8\%$ and standard deviation $1\%.$

Questions

I. Suppose that the yields $of$ the five zero$-$coupon

bonds are all $8\mathrm{.}5\%$ one year from today. What

are the returns $of$ each bond over the period?

Using a simulation with $1000$ iterations, esti$-$

mate the expected return $of$ each bond over

the year. Estimate the standard deviations $of$

the returns.

Comment $on$ the following statement: "The

expected yield $of$ the $30-$year bond one year

from today $is8\%.At$ that yield, its price would

$be$ $$10\mathrm{.}73.$ The return for the year would $be8\%=\frac{10\mathrm{.}73-9\mathrm{.}94}{9\mathrm{.}94}.$ Therefore, the aver$-$

age return for the bond should $be8\%as$ well. $A$

simulation isn't really necessary. Any difference

between $8\%$ and the answer $in$ Question $2$ must

$be$ due $to$ simulation error."