Part $2$ Curve Steepening Trade

For this part, use the Fama$-$Bliss interest rate table. $($Please see data below$).$ The Fama$-$Bliss table contains zero$-$coupon rates $($reported in decimals$).$ Assume for this part that the Fama$-$Bliss rates are annualized using annually compounding. The Fama Bliss file reports decimals $($not percentages$)$

Suppose it is currently Nov. $30,2010$ and you hold a long position in $1-$year zero$-$coupon bonds worth $$1$M in market value. You believe that the yield curve is currently too flat and you think that the curve will steepen soon. However, you are unsure if the general level of interest rates will change $($up$,$ down, or stay the same$).$ You would like to set up a trade that makes money if the curve steepens but is unaffected by level shifts in the yield curve $($hedged against parallel shifts$)$ ie$.$ a $$curve steepening$$ trade.

$1.$ Find the change in the value of the $1-$year bond if the interest rate increases by $1$ basis point

$2.$ Find the change in the value of the $1-$year bond if the interest rate decreases by $1$ basis point

$3.$ Find the change in the value of the $5-$year bond if the interest rate increases by $1$ basis point

$4.$ Find the change in the value of the $5-$year bond if the interest rate decreases by $1$ basis point

$5.$ Find the modified duration of the $1-$year bond.

$6.$ Find the modified duration of the $5-$year bond.

$7.$ Use Modified Duration to determine the total face value in $5-$year zero$-$coupon bonds you would need to buy $(+)$ or short $(-)$ in order to approximately offset the change in portfolio value if the $1-$year and $5-$year interest rates change by the same amount $($parallel shift$).$

$8.$ Find the total value of the portfolio including all $1-$year bonds and all $5-$year bonds

$9.$ Find the current value of the portfolio using the Fama and Bliss rates for December $2010$

$10.$ Suppose that you entered the portfolio positions both long and short $($if applicable$)$ that you determined above. Now, suppose that one month has passed, compute the current $($actual not approximate$)$ value of your portfolio.

$$ The $1-$year bond is now a$($n$)11-$month bond

$$ The $5-$year bond is now a$($n$)4-$year $11-$month bond

$$ For valuation purposes, let$$s assume that the $1-$year zero rate and $5-$year zero rate apply to the $11-$month and $4-$year $11-$month bond, respectively. By how much did the portfolio value change?

$11.$ Explain why the portfolio value did or did not change. Did the value increase or decrease? Why?

Data:

date yield$1$ yield$2$ yield$3$ yield$4$ yield$5$

$6/30/19760\mathrm{.}06460\mathrm{.}0703980\mathrm{.}0731320\mathrm{.}0747320\mathrm{.}076402$

$7/30/19760\mathrm{.}06120\mathrm{.}067610\mathrm{.}0708730\mathrm{.}0731310\mathrm{.}075482$

$8/31/19760\mathrm{.}0590\mathrm{.}0651960\mathrm{.}0675950\mathrm{.}0699160\mathrm{.}072321$

$9/30/19760\mathrm{.}05780\mathrm{.}0634750\mathrm{.}0661980\mathrm{.}0686810\mathrm{.}071253$

$10/29/19760\mathrm{.}0550\mathrm{.}0604610\mathrm{.}0635010\mathrm{.}0659730\mathrm{.}068533$

$11/30/19760\mathrm{.}04950\mathrm{.}0543280\mathrm{.}0571470\mathrm{.}0590580\mathrm{.}061031$

$12/31/19760\mathrm{.}04860\mathrm{.}0535290\mathrm{.}0573030\mathrm{.}0595870\mathrm{.}061948$

$1/31/19770\mathrm{.}05560\mathrm{.}0619930\mathrm{.}065350\mathrm{.}0670940\mathrm{.}06891$

$2/28/19770\mathrm{.}05560\mathrm{.}0609590\mathrm{.}0653950\mathrm{.}0679650\mathrm{.}070636$

$3/31/19770\mathrm{.}0540\mathrm{.}0602840\mathrm{.}0646030\mathrm{.}0673920\mathrm{.}070288$

$4/29/19770\mathrm{.}0560\mathrm{.}0615670\mathrm{.}064930\mathrm{.}067290\mathrm{.}069737$

$5/31/19770\mathrm{.}05860\mathrm{.}0618990\mathrm{.}0651770\mathrm{.}0672150\mathrm{.}069326$

$6/30/19770\mathrm{.}05720\mathrm{.}060910\mathrm{.}0635390\mathrm{.}0655250\mathrm{.}067578$

$7/29/19770\mathrm{.}06220\mathrm{.}0650910\mathrm{.}0673080\mathrm{.}0690340\mathrm{.}070818$

$8/31/19770\mathrm{.}06360\mathrm{.}0659760\mathrm{.}0674490\mathrm{.}0685760\mathrm{.}069742$

$9/30/19770\mathrm{.}06660\mathrm{.}0682550\mathrm{.}0695230\mathrm{.}0704370\mathrm{.}071382$

$10/31/19770\mathrm{.}07050\mathrm{.}0723650\mathrm{.}0735270\mathrm{.}0740540\mathrm{.}074605$

$11/30/19770\mathrm{.}06920\mathrm{.}0711680\mathrm{.}0723270\mathrm{.}0730770\mathrm{.}073855$

$12/30/19770\mathrm{.}06980\mathrm{.}0722870\mathrm{.}0737650\mathrm{.}0747320\mathrm{.}075735$

$1/31/19780\mathrm{.}07260\mathrm{.}0747790\mathrm{.}0758310\mathrm{.}076640\mathrm{.}07748$

$2/28/19780\mathrm{.}07320\mathrm{.}0757950\mathrm{.}0771670\mathrm{.}0780260\mathrm{.}07892$

$3/31/19780\mathrm{.}07480\mathrm{.}0770850\mathrm{.}0784620\mathrm{.}0792110\mathrm{.}079992$

$4/28/19780\mathrm{.}0770\mathrm{.}0798080\mathrm{.}079880\mathrm{.}0804810\mathrm{.}081102$

$5/31/19780\mathrm{.}07940\mathrm{.}0812730\mathrm{.}0821170\mathrm{.}0825410\mathrm{.}082985$

$6/30/19780\mathrm{.}08380\mathrm{.}0848420\mathrm{.}0853720\mathrm{.}0851840\mathrm{.}084999$

$7/31/19780\mathrm{.}08330\mathrm{.}0847590\mathrm{.}084740\mathrm{.}0846730\mathrm{.}084609$

$8/31/19780\mathrm{.}08480\mathrm{.}0845920\mathrm{.}0837260\mathrm{.}0839710\mathrm{.}084213$